submanifolds of product spaces - lirias

Arenberg Doctoral School of Science, Engineering & Technol-ogyFaculty of ScienceDepartment of Mathematics

Submanifolds of product spaces

Daniel KOWALCZYK

Dissertation presented in partialfulfillment of the requirements forthe degree of Doctorin Science

July 2011

Submanifolds of product spaces

Daniel KOWALCZYK

Jury:Prof. dr. L. Verstraelen, chairProf. dr. F. Dillen, promotorProf. dr. P. Igodtdr. J. Van der VekenProf. dr. L. VranckenProf. dr. A.L. Albujer-Brotons(University of Cordoba)

Prof. dr. S. Haesen(Katholieke Hogeschool Kempen and

Hasselt University)

Dissertation presented in partialfulfillment of the requirements forthe degree of Doctorof Science

July 2011

© Katholieke Universiteit Leuven – Faculty of ScienceCelestijnenlaan 200B, B-3001 Heverlee(Belgium)

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Contents

Contents i

1 Preliminaries 11.1 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . 11.2 Riemannian submanifolds . . . . . . . . . . . . . . . . . . 5

2 Fundamental theorem of submanifolds in product spaces 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Riemannian products and submanifolds in Riemannian

products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Riemannian submanifolds of M(c1)×M(c2) . . . . . . . . 142.4 Proofs of the main theorems . . . . . . . . . . . . . . . . . 212.5 Minimal surfaces in M(c1)×M(c2) . . . . . . . . . . . . . 27

3 Constant angle surfaces in product spaces 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Curves in M2(c). . . . . . . . . . . . . . . . . . . . . . . . 333.3 Constant angle surfaces . . . . . . . . . . . . . . . . . . . 35

3.3.1 Complex structures . . . . . . . . . . . . . . . . . . 353.3.2 Totally geodesic surfaces . . . . . . . . . . . . . . . 373.3.3 f is proportional to the identity . . . . . . . . . . . 403.3.4 f is not proportional to the identity . . . . . . . . 41

3.4 Existence results . . . . . . . . . . . . . . . . . . . . . . . 443.5 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . 50

i

ii CONTENTS

4 On extrinsic symmetries of hypersurfaces in product spaces 664.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Extrinsic symmetries . . . . . . . . . . . . . . . . . . . . . 674.3 Warped product metrics . . . . . . . . . . . . . . . . . . . 694.4 Extrinsic symmetries in Hn × R . . . . . . . . . . . . . . . 70

4.4.1 Hypersurfaces of Hn × R . . . . . . . . . . . . . . . 704.4.2 Rotation hypersurfaces in Hn × R . . . . . . . . . 724.4.3 Totally Umbilical hypersurfaces in Hn × R . . . . . 744.4.4 Semi-parallel hypersurfaces of Hn × R . . . . . . . 774.4.5 Parallel hypersurfaces in Hn × R . . . . . . . . . . 80

4.5 Extrinsic symmetries in M(c1)×M(c2) . . . . . . . . . . 814.5.1 Hypersurfaces of M(c1)×M(c2) . . . . . . . . . . 814.5.2 Totally umbilical hypersurfaces . . . . . . . . . . . 824.5.3 Semi-parallel hypersurfaces . . . . . . . . . . . . . 89

5 PMC surfaces in product spaces 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Preliminaries and definitions . . . . . . . . . . . . . . . . . 985.3 Hopf differential for PMC surfaces in M(c1)×M(c2) . . . 995.4 Non-negative PMC surfaces in M(c1)×M(c2) . . . . . . 102

Bibliography 107

Introduction

The research domain of this thesis is the theory of submanifolds. Thetheory of submanifolds is one of the most important subjects of differentialgeometry and is a generalization of the study of curves and surfaces of the3-dimensional Euclidean space E3, which formed the initial developmentof differential geometry in the 18th and 19th century, to any number ofdimensions and codimensions and to arbitrary ambient spaces.

In the past the differential geometers restricted themselves to the studyof submanifolds in the Euclidean space En. There are two reasons forthis. The first reason is the natural character of the Euclidean space.The Euclidean space is namely homogeneous, i.e. the Euclidean space isa space with property that the action of the isometry group is transitive,and isotropic, i.e. every orthonormal frame in the tangent space at apoint can be mapped by an isometry to any other orthonormal frameof the tangent space at that point. The first property tells us that theEuclidean space looks the same at every point and the second propertytells us that the different directions of the Euclidean space have nodistinguished geometric meaning. We can say that the Euclidean spaceis as uniform as possible. The second reason is the simplicity of thecurvature tensor of the Euclidean space En. The curvature tensor ofthe Euclidean space En vanishes and as a consequence we have that theequations of Gauss, Codazzi and Ricci, which can be formulated for moregeneral ambient spaces, are easier to use and to handle.

In the last decade the differential geometers started the study of surfacesin 3-dimensional simply connected homogeneous spaces, i.e. they droppedthe restriction of isotropy. This class of spaces consist of the real spaceforms, i.e. E3, S3 and H3, the other five Thurston geometries (the productspaces S2 × R and H2 × R, the Heisenberg group Nil3, the universal

iii

iv INTRODUCTION

covering of PSL2(R) and the Lie group Sol3, see [55]), 3-dimensionalBerger spheres and Lie groups with left-invariant metrics. We remark thatthe isometry group of the real space forms has the maximal dimension6, the isometry group of the product spaces, the Heisenberg group, theuniversal covering of PSL2(R) and the Berger spheres has dimension4 and the isometry group of the Lie group Sol3 and other Lie groupsequipped with a left-invariant metric has dimension 3. The differentialgeometers first studied surfaces in simply connected homogeneous spaceswith isometry group of dimension 4, because on these homogeneousspaces there exists an Killing vector field whose covariant derivative canbe easily computed, i.e. we have an special direction with a distinguishedgeometric meaning. This Killing vector field allowed the geometers tohandle the surfaces in these homogeneous spaces better. In fact thereexists a necessary and sufficient condition for a surface to be locallyisometrically immersed into a simply connected homogeneous spacewith a 4-dimensional isometry group. This condition is expressed interms of the metric, second fundamental form (like in the case when thehomogeneous space is a real space form) and the data arising form thethe Killing vector field, see [13]. The easiest case to proof this immersiontheorem for 3-dimensional homogeneous spaces is the case of productspaces S2 ×R and H2 ×R. The Killing vector in this case is the parallelvertical vector ∂t. In fact in this case there exist necessary and sufficientconditions to isometrically immerse a n-dimensional Riemannian manifoldinto the product space M(c)× R, where M(c) is an n-dimensional realspace form of curvature c 6= 0.

In this thesis we will study submanifolds of a product manifold M(c1)×M(c2) of two spaces of constant sectional curvature c1 and c2, respectively.We will give now a short overview of the results obtained in this thesis.

Chaper 1 In the first chapter we will give a short survey of Riemannianmanifolds and submanifolds. This chapter is intended to collect all thetheory that will be needed in the following chapters and to establish thenotation for the rest of the thesis.

Chapter 2 The goal of the second chapter is to generalize the theoremof Daniel about the existence and uniqueness of immersion of a n-dimensional Riemannian manifold into the Riemannian productM(c)×R

INTRODUCTION v

as a hypersurface to a theorem about the existence and uniqueness ofimmersion of a n-dimensional Riemannian manifold into the RiemannianproductM(c1)×M(c2) as a submanifold with arbitrary codimension. Wewill therefore first define the notion of a Riemannian product manifold.Let us remark that if the components of M(c1)×M(c2) have dimensionbigger then 1, then we can not define a Killing vector field that isappropriate for the study of submanifolds of M(c1) × M(c2). Wewill solve this problem by re-introducing the product structure F of aRiemannian product manifold. The product structure F of a Riemannianproduct manifold is a (1, 1) tensor field that helps us to control thetangent space of the Riemannian product manifold and is defined asthe reflection with respect to the first component of the tangent spaceof the Riemannian product manifold. The product structure F will bea symmetric parallel (1, 1) tensor field such that F 2 = I. In study ofsubmanifolds of the Riemannian product manifold M(c1) ×M(c2) wewill restrict this product structure to the submanifold and decompose theimage of the product structure into a tangent component and a normalcomponent of the submanifold. This will give us (1, 1) tensor fields definedon the submanifold and will permit us to write the Gauss, Codazzi andRicci equation in terms of these tensor fields defined on the submanifold.These (1, 1) tensor fields will satisfy some compatibility equations thatcan be deduced from the conditions of the product structure F and theformulas of Gauss and Weingarten. We will show that these compatibilityequations together with the equations of Gauss, Codazzi and Ricci aresufficient to isometrically immerse a Riemannian manifold into the theRiemannian product manifold M(c1)×M(c2):

Theorem (see Theorem 2.3.4). Let (M, g) be a simply connected n-dimensional Riemannian manifold, ν a Riemannian vector bundle overM of rank m with metric g, ∇⊥ a connection on ν compatible withthe metric g, σ a symmetric (1, 2) tensor with values in ν and let f :TM → TM , t : ν → ν and h : TM → ν be (1, 1) tensors over M .Define S : ν → End(TM) by g(SξX,Y ) = g(σ(X,Y ), ξ) and define s :ν → TM by g(sξ,X) = g(ξ, hX) for X,Y ∈ TM , ξ ∈ ν. Suppose that(M, g, ν, g, σ,∇⊥, f, h, t) satisfies the compatibility equations for M(c1)×M(c2). Then, there exists an isometric immersion ψ from M intoM(c1)×M(c2) such that σ and S are the second fundamental form andshape operator of M in M(c1) ×M(c2), respectively, ν is isomorphicto the normal bundle of ψ(M) in M(c1) ×M(c2) by an isomorphism

vi INTRODUCTION

ψ : ν → T⊥ψ(M) and such that

F (ψ∗X) = ψ∗(fX) + ψ(hX),

andF (ψξ) = ψ∗(sξ) + ψ(tξ),

where F is the product structure of M(c1)×M(c2).

We will use this theorem to show the existence of a one-parameterfamily of isometric minimal deformations of a given minimal surface inM(c1)×M(c2). We obtain this family of minimal surfaces by rotating theWeingarten endomorphism of the minimal surface. This is an analogue ofthe associate family of a minimal surface in E3. We prove the followingtheorem:

Theorem. Let Σ be a simply connected Riemann surface and x : Σ→M(c1) × M(c2) a conformal minimal immersion. Let T⊥Σ be thenormal bundle of Σ, σ the second fundamental form and ∇⊥ the normalconnection of Σ in M(c1)×M(c2). Let f, h, s and t be (1, 1) tensor fieldsover Σ defined by (2.9) and (2.10). Let p0 ∈ Σ. Then there exists aunique family of minimal conformal immersions xθ : Σ→M(c1)×M(c2)such that:

1. xθ(p0) = x(p0) and (dxθ)p0 = (dx)p0,

2. the metrics induced by xθ and x are the same,

3. the second fundamental form of xθ(Σ) in M(c1)×M(c2) is givenby σθ(X,Y ) = σ(TθX,Y ) for every X,Y ∈ TΣ,

4. F (dxθX) = dxθ(fθX) + hθX and Fξ = dxθsθξ + tθξ for everyX ∈ TΣ and ξ ∈ T⊥Σ.

Moreover we have x0 = x and the family xθ is continuous with respectto θ. The family of immersions (xθ)θ∈R is called the associate familyof immersion x : Σ → M(c1) ×M(c2). The immersion xπ

2is called

the conjugate immersion of x. The immersion xπ is called the oppositeimmersion of x.

In the same chapter we will also formulate the existence and uniquenesstheorem for hypersurfaces in M(c1) × M(c2). We will need thisformulation in the chapter on extrinsic symmetries of hypersurfacesof M(c1)×M(c2).

INTRODUCTION vii

Chapter 3 In this chapter we will generalize the notion of a constantangle surface of M2(c)× R, introduced in [21], to a notion of constantangle surface of M2(c1)×M2(c2). We will therefore use the definitionof angles between two dimensional linear subspaces of R4 given in [60].One can easily deduce then that the definition of a constant anglesurface of M2(c1)×M2(c2) is indeed a generalization of the definition ofconstant angle surface in M2(c)× R. We will show that these constantangle surfaces have constant Gaussian curvature and constant normalcurvature. We will also give an complete classification of constant anglesurfaces of M2(c1) ×M2(c2). Some of the constant angle surfaces ofM2(c1)×M2(c2) will be described with the Sine-Gordon and Sinh-Gordonequation together with their Bäcklund transformation, see [54]. We liketo remark that the definition of constant angle surface ofM2(c1)×M2(c2)can be generalized to a definition of a constant angle submanifold of aRiemannian product manifold M(c1)×M(c2).

Chapter 4 In the fourth chapter we extend the study of Van derVeken and Vrancken on extrinsic symmetries of hypersurfaces of Sn × Rto the study of extrinsic symmetries of hypersurfaces of Hn × R andM(c1)×M(c2) with c1c2 6= 0. We will namely classify totally umbilicalhypersurfaces, parallel hypersurfaces and semi-parallel hypersurfaces ofHn × R and M(c1) ×M(c2). In order to decribe the totally umbilcalhypersurfaces and semi-parallel hypersurfaces of Hn×R we will need thenotion of a revolution hypersurface of Hn×R that was introduced in [20].We will study the extrinsic symmetries of Hn ×R using the formalism ofDaniel [14]. The extrinsic symmetries of hypersurfaces of M(c1)×M(c2)will be studied with the formalism introduced in Chapter 2 of this thesis.

Chapter 5 In the last chapter of the thesis we will investigate parallelmean curvature surfaces of M(c1)×M(c2). i.e. surfaces with a parallelmean curvature vector H. The recent development of the study ofsubmanifolds in homogeneous spaces started with the research of constantmean curvature spheres, i.e. closed oriented surfaces with genus 0,of M2(c) × R by Abresch and Rosenberg. In [1], they constructeda holomorphic differential on a constant mean curvature surfaces ofM2(c)×R, which is a perturbation of the classical holomorphic differentialon CMC surface of 3-dimensional space forms constructed by Hopf in[32]. Using this holomorphic differential they are able to classify constant

viii INTRODUCTION

mean curvature spheres of M2(c) × R. In [4], the authors generalizethe results of Abresch and Rosenberg to higher codimensional parallelmean curvature surfaces of M(c) × R. In this thesis we will constructa holomorphic differential for parallel mean surfaces of M(c1)×M(c2).We will first define a quadratic form Qc1,c2 by

Qc1,c2(X,Y ) = 2|H|2g(σ(X,Y ), H) + c1g((P1X ∧ P1H)H,Y )

+ c2g((P2X ∧ P2H)H,Y ),

where σ is the second fundamental form of the parallel mean curvaturesurface and P1 and P2 are defined by I+F

2 and I−F2 , respectively. Then

we will show that the (2, 0)-part of the quadratic form is a holomorphicfunction.

Theorem. LetM2 be a parallel mean curvature surface ofM(c1)×M(c2).Then the (2, 0)−part Q(2,0)

c1,c2 of the quadratic form Qc1,c2 is holomorphic.

We will also show that the quadratic form Qc1,c2 can be used to deducea Simons’ type formula for parallel mean curvature surfaces of M(c1)×M(c2). This will be a generalization of a result proven by Fetcu andRosenberg [30]. We will show that this Simons’ type formula can be usedin the study of parallel mean curvature surfaces of M(c1)×M(c2) withnon-negative sectional curvature.

Chapter 1

Preliminaries

In this first chapter we will give a short survey of Riemannian manifoldsand submanifolds in order to establish the notations. We refer to [52]for more detailed information on these topics.

1.1 Riemannian manifolds

Let M be a topological space. We assume also that M satisfies theHausdorff axiom. An open chart on M is a pair (U, φ), where U is anopen subset of M and φ is a homeomorphism of U onto an open subsetof Rn.

Definition 1.1.1. Let M be a topological space which satisfies theHausdorff axiom. A differential structure on M of dimension n isa collection of open charts (Ui, φi)i∈I on M such that the followingconditions are satisfied:

1. M =⋃i∈I Ui;

2. For each pair i, j ∈ I the mapping φj ◦ φ−1i is a differentiable

mapping of φi(Ui ∩ Uj) onto φj(Ui ∩ Uj);

3. The collection (Ui, φi) is a maximal family of open charts for whichthe previous two conditions hold.

A differentiable manifold (or simply manifold) of dimension n is atopological Hausdorff space with a differentiable structure of dimensionn.

1

2 PRELIMINARIES

Given a map f : M → N between two manifolds M and N , f is said tobe differentiable, if for every chart (Ui, φ) of M and every chart (Vj , ψj)of N such that f(Ui) ⊂ Vj , the mapping ψj ◦ f ◦ φ−1

i of φi(Ui) intoψj(Uj) is differentiable. By a differentiable curve in M , we shall mean adifferentiable mapping of an open interval of R into M . We use now thedefinition of a curve to define the tangent space of M at a point.

Definition 1.1.2. A tangent vector at a point p ∈M is an equivalenceclass of differentiable curves c : ]−ε, ε[ → M with c(0) = p, wherec ∼ c⇔ (φ ◦ c)′(0) = (φ ◦ c)′(0) for every open chart (U, φ) containing p.

The set of tangent vectors of M at p is called the tangent space of Mat p and is denoted by TpM . One can show that the tangent spaceTpM is a n-dimensional real vector space. We call the union

⋃p∈M TpM

the tangent bundle and denote the tangent bundle by TM . One canshow that TM has a natural differentiable structure and hence we havethat TM is a manifold. A vector field X on M is an assignment of avector Xp in TpM to each point p of M , i.e. X is a mapping from Mto TM . X is called differentiable, if X is differentiable as a mappingbetween the two manifolds M and TM . In the following we will denotethe set of all differentiable vector fields on M also by TM , the meaningof TM will be clear from the context. There exists also a more algebraicdefinition of a tangent vector and hence of a vector field. Namely, onecan define tangent vectors as derivations acting on scalar functions ofM , i.e. a tangent vector is a mapping from the set of scalar mappingsof M to R that satisfies R-linearity and the product rule. If X and Yare vector fields, define the bracket [X,Y ] as a mapping from the ring ofdifferentiable functions on M onto itself by

[X,Y ]f = X(Y f)− Y (Xf).

One can show that [X,Y ] acts as a derivation and hence [X,Y ] is also avector field. We call [X,Y ] the Lie-bracket of X and Y .

A mapping f : N → M is called an immersion if the differential (f∗)pat p is injective for every point p of N . N is then called an immersedsubmanifold of M . If the immersion f is injective, then f is called animbedding of N into M . We give now the definition of a distributionon a manifold. A distribution S of dimension r on M is an assignmentto each point p of M an r-dimensional subspace Sp of TpM . We call S

RIEMANNIAN MANIFOLDS 3

differentiable if every point p has an neighbourhood U and r differentiablevector fields which form a basis of Sq at every q ∈ U . A vector fieldis said to belong to S if Xp ∈ Sp for all p ∈ M . We call S involutiveif [X,Y ] ∈ S for any X,Y ∈ S. We shall always assume that thedistribution is differentiable. A connected manifold N of M is called anintegral manifold of S if f∗(TpN) = Sf(p) for every p ∈ N , where f isan imbedding of N into M . If there is no other integral manifold of Swhich contains N , then N is called a maximal integral manifold of S.We now formulate the theorem of Frobenius as follows.

Proposition 1.1.3. Let S be an involutive distribution on a manifoldM . Through every point p ∈M , there passes a unique maximal integralmanifold N(p) of S. Any integral manifold of S through p is an opensubmanifold of N(p).

We define now a derivative of a vector field on a manifold with the notionof connection.

Definition 1.1.4. A connection on a manifold is a rule ∇ which assignsto each X ∈ TM a linear mapping ∇X of the vector space TM into itselfsatisfying the following two conditions:

∇fX+gY = f∇X + g∇Y ,

∇X(fY ) = f∇XY +X(f)Y,

for every scalar function f and g on M and X,Y ∈ TM . The operator∇X is called the covariant differentiation with respect to X.

We define now a Riemannian metric on a manifold M . This is a tensorfield g of type (0, 2), for definition of tensor field see for example [52],which satisfies the following two conditions:

1. It is symmetric: g(X,Y ) = g(Y,X) for any X,Y ∈ TM ;

2. It is positive definite: g(X,X) ≥ 0 for every X ∈ TM andg(X,X) = 0 if and only if X = 0.

A manifold M with a Riemannian metric g is called a Riemannianmanifold and is denoted as the pair (M, g). We will denote the metricg sometimes by 〈 , 〉. The Riemannian metric gives rise to an inner

4 PRELIMINARIES

product on each tangent space TpM at p and to a metric on the manifoldM . We call a map F : M →M ′ between the two Riemannian manifolds(M, g) and (M ′, g′) an isometry if F is a diffeomorphism which preservesthe metric, i.e. F ∗g′ = g, we say that M and M ′ are isometric.

We can always find a connection on a Riemannian manifold (M, g) thatis compatible with the metric g and the differentiable structure of themanifold M . Moreover this connection is unique and is called the Levi-Civita connection of (M, g). More precisely, we have:

Theorem 1.1.5. On a Riemannian manifold (M, g) there exists oneand only one connection satisfying the following conditions:

1. ∇XY −∇YX = [X,Y ], ∇ is torsion free;

2. Xg(Y,Z) = g(∇XY,Z) + g(Y,∇XZ), g is parallel;

for every X,Y, Z ∈ TM .

Using the Levi-Civita connection ∇, we can define the curvature tensorR as a (1, 3) tensor field on (M, g) by

R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z

for every X,Y, Z ∈ TM . For each plane π in the tangent space TpM ,the sectional curvature K(p, π) is defined by

K(p, π) = g(R(v, w)w, v),

where {v, w} is an orthonormal basis of π. One can show that K(p, π) isindependent of the orthonormal basis. One can also deduce from somealgebraic properties of the curvature tensor that the set of values K(p, π)for all planes π of TpM determines the Riemannian curvature tensorof (M, g) at p. If K(p, π) is independent of the plane π of TpM andindependent of the point p of M , then M is called a space of constantcurvature. The curvature tensor of a space of constant curvature is givenby

R(X,Y )Z = c(X ∧ Y )Z = c (g(Y,Z)X − g(X,Z)Y ) .A Riemannian manifold is called a real space formM(c) if the Riemannianmanifold M(c) is simply connected, complete and has constant sectionalcurvature. A real space form M(c) is isometric with the Euclidean spaceEn if c = 0, with the sphere Sn(c) of radius 1√

cif c > 0 or with the

hyperbolic space Hn(c) if c < 0.

RIEMANNIAN SUBMANIFOLDS 5

1.2 Riemannian submanifolds

Let (M, g) and (M, g) be Riemannian manifolds of dimension n andn + m, respectively, then we call an immersion f : M → M anisometric immersion if f∗g = g. The pair (M,f) is called a Riemanniansubmanifold of M and m is the codimension of M in M . A mappingwhich associates to each point p of M a tangent vector to M at f(p) iscalled a vector field along f . Since (f∗)p is an injective linear map atevery point p of M , we have that (f∗)p maps the n-dimensional tangentspace to M at p, to a n-dimensional subspace of the tangent space to Mat f(p) for every point p of M . We will identify this subspace with thetangent space TpM to M at the point p, i.e. we will identify v ∈ TpMwith (f∗)p(v) ∈ f∗(TpM). Moreover we will identify the metric g withf∗g and denote both metrics by g or 〈 , 〉. A vector field ξ is called anormal vector field to M if 〈ξ,X〉 = 0 for all tangent vector fields X toM . The normal space at a point p of M will be denoted by T⊥p M andthe normal bundle T⊥M is the union of the normal spaces to M . It iseasy to see that vector fields along f can be decomposed in a tangentvector field to M and a normal vector field to M .

Let us denote the Levi-Civita connections of M and M by ∇ and ∇,respectively . We can decompose the vector field ∇XY along f , whereX,Y are vector fields on M , in a tangent and a normal component. Weobtain in this way the formula of Gauss:

∇XY = ∇XY + σ(X,Y ),

where σ is a symmetric (1, 2) tensor field which takes values in the normalbundle T⊥M . σ is called the second fundamental form of M in M . Weobtain the formula of Weingarten by decomposing ∇Xξ in a tangent anda normal component, where ξ is a normal vector field on M :

∇Xξ = −SξX +∇⊥Xξ,

where Sξ is a (1, 1) tensor field on M , which is called the shape operatorassociated to ξ, and∇⊥ is a connection in the normal bundle T⊥M , whichis called the normal connection of M in M . The second fundamentalform σ and the shape operator Sξ are related by the formula

g(σ(X,Y ), ξ) = g(SξX,Y ).

6 PRELIMINARIES

Let us denote the curvature tensors of ∇, ∇ and ∇⊥ by R, R and R⊥,respectively, and by > and ⊥ the projection of Tf(p)M on TpM and T⊥p M ,respectively. Deriving now the formulas of Gauss and Weingarten, weobtain the equations of Gauss, Codazzi and Ricci. They are respectivelygiven by

>(R(X,Y )Z) = R(X,Y )Z + Sσ(X,Z)Y − Sσ(Y,Z)X,

⊥(R(X,Y )Z) = (∇σ)(X,Y, Z)− (∇σ)(Y,X,Z),

⊥(R(X,Y )ξ) = R⊥(X,Y )ξ + σ(SξX,Y )− σ(X,SξY ),

where X,Y, Z are tangent vector fields of M , ξ is a normal vector fieldon M and ∇σ is defined by

(∇σ)(X,Y, Z) = ∇⊥Xσ(Y,Z)− σ(∇XY,Z)− σ(X,∇XZ).

In some chapters we will consider only hypersurfaces, i.e. submanifoldsof codimension one. Let M be a hypersurface of M . We will denote thescalar valued second fundamental form by h. Let N be a unit normalvector field on M with associated shape operator S. Then we defineh(X,Y ) = 〈SX, Y 〉 for X,Y tangent to M . The formulas of Gauss andWeingarten are then given by

∇XY = ∇XY + h(X,Y )N,

∇Xξ = −SX.

The equations of Gauss and Codazzi are given by

>(R(X,Y )Z) = R(X,Y )Z − (SX ∧ SY )Z,

>(R(X,Y )N) = (∇Y S)X − (∇XS)Y,

where (∇XS)Y = ∇XSY − S(∇XY ).

Chapter 2

Fundamental theorem ofsubmanifolds in productspaces

2.1 Introduction

In the second chapter we have seen that the metric g and the secondfundamental form h of a hypersurface M of a Riemannian manifold Msatisfy two equations, namely the Gauss and Codazzi equation. In thecase that M is a space form M(c) of constant sectional curvature c, theseequations become:

R(X,Y )Z = c(X ∧ Y )Z + (SX ∧ SY )Z, (2.1)

(∇XS)Y = (∇Y S)X. (2.2)

So we see that the Gauss and Codazzi equation ((2.1) and (2.2)) involveonly the metric g of the hypersurface M and the second fundamentalform h of M in M(c). We can say that the equations of Gauss andCodazzi are intrinsically in the case that the ambient space is a spaceform, as soon we know the shape operator S. Moreover we have in thiscase that the two equations (2.1) and (2.2) are sufficient conditions for an-dimensional simply connected Riemannian manifold to be isometricallyimmersed in M(c) with given metric g and second fundamental form h:

7

8 FUNDAMENTAL THEOREM OF SUBMANIFOLDS IN PRODUCT SPACES

Theorem 2.1.1 ([52]). Let (M, g) be a simply connected n-dimensionalRiemannian manifold, with Levi-Civita connection ∇ and curvaturetensor R, and let S be a (1, 1) symmetric tensor field on M . Supposethat S satisfies equations (2.1) and (2.2). Then there exists an isometricimmersion x : M → M(c) such that S is the shape operator associatedto some unit normal field N of x(M) in M(c).

If M is an arbitrary Riemannian manifold, then we don’t have that theGauss equation and Codazzi equation are intrinsically defined, becausethe two equations contain a term that involves the curvature tensor ofthe ambient space M . In [14], Daniel showed that the Gauss equationand Codazzi equation of a hypersurface M in M(c) × R are known assoon as we know the projection T of the vertical vector field ∂t on thetangent space of the hypersurface and the normal component ν = 〈N, ∂t〉of ∂t, where N is a unit normal of M in M(c)×R. In this case the Gaussand Codazzi equation are given in terms of T and ν as follows:

R(X,Y )Z = (SX ∧ SY )Z + c ((X ∧ Y )Z + 〈Y, T 〉(T ∧X)Z

− 〈X,T 〉(T ∧ Y )Z)

(∇Y S)X − (∇XS)Y = cν(X ∧ Y )T,

where X,Y, Z are vector fields tangent to M and S is the shape operatorassociated to N . Moreover, by using the fact that ∂t is a parallel vectorfield in M(c)× R, i.e. ∇X∂t = 0 for every X ∈ T (M(c)× R), where ∇is the connection of the ambient space M(c) × R, and the formulas ofGauss and Weingarten, we obtain

∇XT = νSX, (2.3)

X(ν) = −〈SX, T 〉. (2.4)

Daniel showed in [14] that the equations of Gauss and Codazzi forhypersurfaces in M(c)× R together with equations (2.3) and (2.4) aresufficient and necessary to isometrically immerse a Riemannian manifoldin M(c)× R:

Theorem 2.1.2 ([14]). Let (M, g) be a simply connected n-dimensionalRiemannian manifold, with Levi-Civita connection ∇ and curvaturetensor R, and let S be a symmetric (1, 1) tensor field on M . Let T and

INTRODUCTION 9

ν be a vector field and a smooth function on M such that ‖T‖2 + ν2 = 1.Assume that the equations of Gauss and Codazzi for hypersurfaces inM(c) × R and the compatibility equations (2.3) and (2.4) are satisfied.Then there exists an isometric immersion x : M →M(c)× R with unitnormal N , such that the shape operator with respect to this normal isgiven by S and such that ∂t = T + νN . Moreover, the immersion isunique up to global isometries of M(c)× R preserving the orientationsof both M(c) and R.

Our aim in this chapter is to generalize theorem 2.1.2 to the case of theRiemannian product of two space forms M(c1) and M(c2) of constantsectional curvatures c1 and c2, respectively. We give necessary andsufficient conditions for the existence and uniqueness (modulo isometriesof the ambient space) of a isometric immersion of a Riemannian manifoldinto the Riemannian product of two space forms. We note that similarresults were independently proved by Lira, Tojeiro and Vitório in [40].Moreover, it was recently shown by Piccione and Tausk in [46] thatexistence and uniqueness of immersion theorems can be extended toRiemannian manifolds that are "sufficiently homogeneous". This is acondition that can be formulated in terms of the G-structure on themanifold.

We organize the next chapter as follows. In section 2.2 we discuss theRiemannian product manifolds, the product structure of a Riemannianproduct manifold and submanifolds and hypersurfaces of productspaces. We establish the compatibility equations for submanifolds andhypersurfaces in product spaces. In section 2.3 we will discuss isometricimmersions into the Riemannian product M(c1)×M(c2) of two spaceforms and state the main theorems about existence and uniqueness ofan isometric immersion of a Riemannian manifold into the Riemannianproduct M(c1)×M(c2) of two space forms. In Section 2.4 we will proofthe main theorems. We end this chapter with section 2.5, where wediscuss a application of the fundamental theorem. Some parts of thischapter are based on [37].


2.2 Riemannian products and submanifolds inRiemannian products

Riemannian products Let M1 and M2 be two differentiable man-ifolds of dimensions n1 and n2, respectively. Consider the productmanifold M1 ×M2 with natural projections π1 : M1 ×M2 → M1 andπ2 : M1×M2 →M2. The tangent space T(p1,p2)(M1×M2) ofM1×M2 at(p1, p2) is isomorphic to the direct sum Tp1M1

⊕Tp2M2 for every point

(p1, p2) ofM1×M2, i.e. we can identify the vector space T(p1,p2)(M1×M2)with the vector space Tp1M1

⊕Tp2M2. At every point (p1, p2) one

can define the linear endomorphism F(p1,p2) : T(p1,p2)(M1 × M2) →T(p1,p2)(M1 × M2) : (v1, v2) 7→ (v1,−v2). Since F(p1,p2) is defined atevery point of (p1, p2), we can define a field of endomorphisms of thetangent spaces to M1 ×M2, i.e. a (1, 1) tensor field, by

F (X1, X2) = (X1,−X2), (2.5)

whereby X1 ∈ TM1 and X2 ∈ TM2. Notice that F 2 = I, whereby Idenotes the identity transformation on T (M1 ×M2). It is easy to seethat a tangent vector v to M1 × M2 at (p1, p2) lies in Tp1M1 if andonly if Fv = v and that a tangent vector w to M1 ×M2 at (p1, p2) liesin Tp2M2 if and only if Fw = −w. We remark that tr(F ) = n1 − n2,where tr stands for trace, and that the rank of 1

2(I + F ) and 12(I − F ) is

respectively n1 and n2.

If (M1, g1) and (M2, g2) are Riemannian manifolds, consider theRiemannian product metric g on M1 ×M2:

g((X1, X2), (Y1, Y2)) = g1(X1, Y1) + g2(X2, Y2),

for all X1, Y1 ∈ TM1 and for all X2, Y2 ∈ TM2. A product manifold ofRiemannian manifolds equipped with the Riemannian product metric iscalled a Riemannian product manifold or simply Riemannian product.We will call sometimes Riemannian products product spaces. Notice thatthe subspaces Tp1M1 and Tp2M2 of T(p1,p2)(M1×M2) are orthogonal andthat the (1, 1) tensor field F on M1 ×M2, defined by equation (2.5), is asymmetric (1, 1)-tensor on M1 ×M2 with respect to g. Denote by ∇ theLevi-Civita connection of (M1 ×M2, g). It can be shown that

(∇XF )(Y ) = 0,

RIEMANNIAN PRODUCTS AND SUBMANIFOLDS IN RIEMANNIAN PRODUCTS 11

i.e.∇X(FY ) = F (∇XY ).

We summarize the properties of the (1, 1) tensor field F in the followingproposition:

Proposition 2.2.1. Let (M1×M2, g) be a Riemannian product manifoldwith Levi-Civita connection ∇ and F the (1, 1) tensor field on M1 ×M2,defined by equation (2.5). Then F satisfies the following properties:

F 2 = I, (2.6)

g(FX, Y ) = g(X,FY ), i.e. F is symmetric with respect to g, (2.7)

(∇XF )Y = 0, i.e. F is parallel, (2.8)

for every vector field X,Y of M1 ×M2. We call F the product structureassociated to the Riemannian product manifold M1 ×M2.

Submanifolds in Riemannian products LetM be a n-dimensionalRiemannian submanifold of the Riemannian product manifold M1 ×M2with metric g, second fundamental form σ and normal connection ∇⊥.For any vector field X tangent to M we put

FX = fX + hX, (2.9)

where fX is the tangential part of FX and hX is the normal part ofFX. We remark that f : TM → TM and h : TM → T⊥M are (1, 1)tensor fields over M . For any normal vector field ξ to M we put

Fξ = sξ + tξ, (2.10)

where sξ is the tangential part of Fξ and tξ is the normal part of Fξ.We also remark here that s : T⊥M → TM and t : T⊥M → T⊥M are(1, 1) tensor fields on M . We have now, using equation (2.6), that:

f2 = I − sh, (2.11)

t2 = I − hs, (2.12)

fs+ st = 0, (2.13)

hf + th = 0. (2.14)


Using equation (2.7), we can easily see that:

g(fX, Y ) = g(X, fY ), i.e. f is symmetric with respect to g, (2.15)

g(tξ, η) = g(ξ, tη), i.e. t is symmetric with respect to g, (2.16)

g(sξ,X) = g(ξ, hX), i.e. h is the transpose of s and vise versa. (2.17)

Using equations (2.9) and (2.10) together with the formulas of Gaussand Weingarten, we obtain the following proposition:

Proposition 2.2.2. For X,Y ∈ TM and ξ ∈ T⊥M we have

(∇Xf)(Y ) = ShYX + s(σ(X,Y )), (2.18)

∇⊥XhY − h(∇XY ) = t(σ(X,Y ))− σ(X, fY ), (2.19)

∇⊥Xtξ − t(∇⊥Xξ) = −σ(sξ,X)− h(SξX) and (2.20)

∇Xsξ − s(∇⊥Xξ) = −fSξX + StξX. (2.21)

Proof. We have (∇XF )Y = 0 and (∇XF )ξ = 0 for every X,Y ∈ TMand ξ ∈ T⊥M , because F is parallel. Thus we get, by using formulas ofGauss and Weingarten together with equations (2.9) and (2.10):

0 = ∇XFY − F (∇XY )

= ∇XfY + ∇XhY − F (∇XY )− F (σ(X,Y ))

= ∇XfY + σ(X, fY )− ShYX +∇⊥XhY

− f(∇XY )− h(∇XY )− s(σ(X,Y ))− t(σ(X,Y ))

(2.22)

and0 = ∇XFξ − F (∇Xξ)

= ∇Xsξ + ∇Xtξ + F (SξX)− F (∇⊥Xξ)

= ∇Xsξ + σ(X, sξ)− StξX +∇⊥Xtξ

fSξX + hSξX − s∇⊥Xξ − t∇⊥Xξ.

(2.23)

Taking the tangential and normal parts of the equations (2.22) and (2.23),we obtain equations (2.18), (2.19), (2.20) and (2.21).

RIEMANNIAN PRODUCTS AND SUBMANIFOLDS IN RIEMANNIAN PRODUCTS 13

We will need these equations in order to state the fundamental theoremof submanifolds in product spaces.

To end this section, we will give analogous equations for hypersurfacesimmersed in M1 × M2, because in one of the next chapters we willneed these specific equations. Hence suppose we have a hypersurfaceM isometrically immersed into the Riemannian product M1 ×M2 ofdimension n1 + n2 = n + 1. We denote by ξ the unit normal of M inM1 ×M2 and by S the shape operator associated to ξ. We can put now

FX = fX + u(X)ξ, (2.24)

Fξ = U + λξ, (2.25)

where f, u, U and λ define a linear endomorphism of the tangent bundle,a 1-form, a vector field and a function onM , respectively. Using equation(2.6), we can easily deduce that

f2X = X − u(X)U, (2.26)

u(fX) = −λu(X), (2.27)

fU = −λU, (2.28)

u(U) + λ2 = 1, (2.29)

for every X ∈ TM . We obtain now, using equation (2.7),

g(fX, Y ) = g(X, fY ), i.e. f is symmetric with respect to g,

u(X) = g(U,X), (2.30)

for every X,Y ∈ TM . Furthermore we have the following propositionthat can be proved analogously as Proposition 2.2.2.

Proposition 2.2.3. Let M be a hypersurface in M1 ×M2 and f, u, Uand λ defined as above, then

(∇Xf)Y = g(SX, Y )U + u(Y )SX, (2.31)

(∇Xu)Y = λg(SX, Y )− g(SfX, Y ), (2.32)

∇XU = −fSX + λSX, (2.33)

Xλ = −2u(SX). (2.34)


Remark 2.2.4. Let us remark that equations (2.32) and (2.33) areequivalent, because of equation (2.30).

2.3 Riemannian submanifolds of M(c1)×M(c2)

In this section we will consider submanifolds of the Riemannian productmanifold M(c1) ×M(c2), where M(c1) and M(c2) are space forms ofconstant curvature c1 ∈ R0 and c2 ∈ R, respectively. Denote by n1 andn2 the dimensions of M(c1) and M(c2), respectively.

Denote by Ep = (Rp, 〈., .〉 = dx20 + dx2

1 + · · · + dx2p−1)) and Lp =

(Rp, 〈., .〉1 = −dx20 + dx2

1 + · · · + dx2p−1) the Euclidean and Minkowski

space of dimension p, where (x0, . . . , xp−1) are the standard coordinatesof Rp. We will use the following representations of the space forms M(c)of constant curvature c. If c = 0, then we know that M(c) is isometricto the Euclidean space En. For c > 0 one can show that the space formM(c) is isometric to the n-dimensional sphere Sn(c) of radius 1√

cin En+1,

Sn(c) ={x ∈ Rn+1 | 〈x, x〉 = 1

c

},

with Riemannian metric induced from the ordinary metric 〈., .〉 of En+1.If c < 0 we will represent M(c) by the quadric hypersurface Hn(c) inLn+1,

Hn(c) ={x ∈ Rn+1 | 〈x, x〉1 = 1

c, x0 > 0

},

with the Riemannian metric induced from the Minkowski metric 〈., .〉1 ofLn+1. We define M(c1) ×M(c2) as follows. Suppose first that c2 = 0,then we define M(c1)× En2 as a hypersurface of En1+n2+1 if c1 > 0 oras the hypersurface of Ln1+n2+1 if c1 < 0 by,

M(c1)× En2 = {(x0, . . . , xn1 , x1, . . . , xn2) ∈ Rn1+n2+1 |

sgn(c1)x20 + · · ·+ x2

n1 = 1c1

and x0 > 0 if c1 < 0},

equipped with the induced metric. Then M(c1)×En2 is the Riemannianproduct of the n1−dimensional space form of constant curvature c1 6= 0and the n2−dimensional Euclidean space. The Riemannian product

RIEMANNIAN SUBMANIFOLDS OF M(C1)×M(C2) 15

manifold M(c1) × En2 is a hypersurface of En1+n2+1 or Ln1+n2+1,depending on the sign of c1, with normal vector ξ = (x0, . . . , xn1 , 0, . . . , 0).Using the formula of Gauss for isometric immersions into En1+n2+1 orLn1+n2+1, we find that the Levi-Civita connection ∇ of M(c1)× En2 isgiven by the next expression in terms of the Levi-Civita connection Dof En1+n2+1 or Ln1+n2+1 and the (1, 1) tensor field F of M(c1) × En2

defined by (2.5):

∇XY = DXY + c1

⟨X + FX

2 , Y

⟩(1)ξ.

Consequently, we obtain that the curvature tensor R associated to ∇ isgiven by

R(X,Y )Z = c1

(X + FX

2 ∧ Y + FY

2

)Z.

Suppose now that c2 6= 0. We define M(c1)×M(c2) as a submanifold ofcodimension 2 of En1+n2+2 if c1, c2 > 0, as a submanifold of codimension2 of Ln1+n2+2 if c1c2 < 0 or as submanifold of codimension 2 of Rn1+n2+2

2if c1, c2 < 0, where Rn1+n2+2

2 = (Rn1+n2+2,−dx20 + dx2

1 + · · · + dx2n1 −

dx20 + dx2

1 + · · ·+ dx2n2) by,

M(c1)×M(c2) = {(x0, . . . , xn1 , x0, . . . , xn2) ∈ Rn1+n2+2 |

sgn(c1)x20 + · · ·+ x2

n1 = 1c1, sgn(c2)x2

0 + · · ·+ x2n1 = 1

c2,

x0 > 0 if c1 < 0 and x0 > 0 if c2 < 0},

equipped with the induced metric. Then M(c1) × M(c2) is theRiemannian product of the n1-dimensional space form M(c1) of constantcurvature c1 and the n2-dimensional space form M(c2) of constantcurvature c2. The Riemannian product manifold is a submanifold ofcodimension two of En1+n2+2, Ln1+n2+2 or Rn1+n2+2

2 , depending on thesigns of c1 and c2, with normal vectors ξ = (x0, . . . , xn1 , 0, . . . , 0) andξ = (0, . . . , 0, x0, . . . , xn2). Using the formula of Gauss for isometricimmersions into En1+n2+2, Ln1+n2+ or Rn1+n2+2

2 , we find that the Levi-Civita connection ∇ of M(c1)×M(c2) is given by the next expression interms of the Levi-Civita connectionD of En1+n2+2, Ln1+n2+2 or Rn1+n2+2

2and the (1, 1) tensor field F of M(c1)×M(c2) defined by (2.5):

∇XY = DXY + c1

⟨X + FX

2 , Y

⟩ξ + c2

⟨X − FX

2 , Y

⟩ξ (2.35)


Hence, we obtain that the curvature tensor R associated to ∇ is givenby

R(X,Y )Z = c1

(X + FX

2 ∧ Y + FY

2

)Z+c2

(X − FX

2 ∧ Y − FY2

)Z.

(2.36)Let M be an n-dimensional Riemannian manifold immersed in M(c1)×M(c2). Denote by ∇ the Levi-Civita connection of M , by R thecurvature tensor associated to ∇, by S (respectively σ) the Weingartenendomorphism (respectively the second fundamental form) of M inM(c1) ×M(c2), by ∇⊥ the normal connection of the normal bundleT⊥M of M in M(c1)×M(c2), by R⊥ the curvature tensor associated to∇⊥ and by ∇σ the covariant derivative of σ. The tangent vector fieldswill be denoted by X,Y, Z, . . . and normal vector fields by ξ, η, . . . . Letf, h, s and t be (1, 1) tensor fields as defined in the previous section.The (1, 1) tensor field F of M(c1)×M(c2) defined by (2.5) is symmetricand satisfies F 2 = I, hence we obtain equations (2.11)− (2.17). One canalso easily deduce that the (1, 1) tensor field F of M(c1)×M(c2) definedby (2.5) satisfies equation (2.8) and hence the equations (2.18)− (2.21)are satisfied. Using these notations and equation (2.36) for the curvaturetensor of M(c1)×M(c2), we get that the equations of Gauss, Codazziand Ricci for submanifolds of M(c1)×M(c2) reduce to

R(X,Y )Z = Sσ(Y,Z)X − Sσ(X,Z)Y + c1

(X + fX

2 ∧ Y + fY

2

)Z,

+ c2

(X − fX

2 ∧ Y − fY2

)Z

(∇σ)(X,Y, Z)− (∇σ)(Y,X,Z) = h

(c1(X ∧ Y )Z + fZ

4

− c2(X ∧ Y )Z − fZ4

),

R⊥(X,Y )ξ = c1 + c24 (hX ∧ hY ) ξ − σ(SξX,Y ) + σ(SξY,X).

Remark 2.3.1. We remark that the (1, 1) tensor s is, in some sense, a kindof transpose of the (1, 1) tensor h, because of equation (2.17). Moreover,


we can easily see that equations (2.19) and (2.21) are equivalent becauseof equation (2.17). Analogously we can see that the two equations (2.13)and (2.14) are equivalent. We would also like to remark that the tensort is not uniquely determined by f and h, but a part of it is uniquelydetermined. From equation (2.14), we obtain that the image of h isglobally invariant by t and that the tensor t is uniquely determined onthe image of h. Moreover we have that t2 = I on the kernel of s, whichis equal to the orthogonal complement of the image of h in T⊥M . Sincet is symmetric, this implies that the restriction of t to the kernel of sis diagonalisable with eigenvalues ±1. Hence the only freedom for t isthe choice of the eigenspace for the value 1, whose dimension is given.We like to illustrate these facts with the following example. Considerthe case where M = Σ× {0} ⊂M3(c)×R, where Σ is a totally geodesicsurface of M3(c). It is easy to see that one can take different tensors tthat satisfy the above equations. The previous calculations suggest tointroduce the following definition.

Definition 2.3.2. Let (M, g) be an n-dimensional Riemannian manifold,ν a Riemannian vector bundle over M of rank m with metric g, ∇⊥a connection on ν compatible with the metric g, σ a symmetric (1, 2)tensor with values in ν and let f : TM → TM , t : ν → ν andh : TM → ν be (1, 1) tensors over M . Define S : ν → End(TM)by g(SξX,Y ) = g(σ(X,Y ), ξ) and define s : ν → TM by g(sξ,X) =g(ξ, hX) for X,Y ∈ TM , ξ ∈ ν. We say that (M, g, ν, g, σ,∇⊥, f, h, t)satisfies the compatibility equations for M(c1)×M(c2) if

f is a symmetric (1, 1) tensor field on M such that f2X = X − shX,(2.37)

t is a symmetric (1, 1) tensor field on M such that t2ξ = ξ − hsξ,(2.38)

hfX + thX = 0, (2.39)

the rank of the bundle map I+F2 and I−F

2 is respectively n1 and n2, withn1 + n2 = n + m, where F : TM

⊕ν → TM

⊕ν is the bundle map

defined by FX = fX + hX and Fξ = sξ + tξ for every X ∈ TM and


ξ ∈ ν and, for all X,Y, Z ∈ TM and all ξ ∈ ν:

R(X,Y )Z = Sσ(Y,Z)X − Sσ(X,Z)Y + c1

(X + fX

2 ∧ Y + fY

2

)Z,

+ c2

(X − fX

2 ∧ Y − fY2

)Z, (2.40)

(∇σ)(X,Y, Z)− (∇σ)(Y,X,Z) = h

(c1(X ∧ Y )Z + fZ

4

− c2(X ∧ Y )Z − fZ4

), (2.41)

R⊥(X,Y )ξ = c1 + c24 (hX ∧ hY ) ξ − σ(SξX,Y ) + σ(SξY,X). (2.42)

(∇Xf)(Y ) = ShYX + s(σ(X,Y )), (2.43)

∇⊥XhY − h(∇XY ) = t(σ(X,Y ))− σ(X, fY ), (2.44)

∇⊥Xtξ − t(∇⊥Xξ) = −σ(sξ,X)− h(SξX). (2.45)

Theorem 2.3.3. Let ψ : M → M(c1) × M(c2), resp. ψ′ : M →M(c1) × M(c2), be isometric immersions, with corresponding secondfundamental form σ, resp. σ′, shape operator S, resp. S′, normal spaceT⊥M , resp. T⊥′M . Let f and h be (1, 1) tensors on M defined by (2.46)and f ′ and h′ similarly for ψ′. Suppose that the following conditionshold:

1. fX = f ′X for every X ∈ TpM and p ∈M .

2. There exists an isometric bundle map φ : T⊥M → T⊥′M such that

φ(σ(X,Y )) = σ′(X,Y ),

φ(∇⊥Xξ) = ∇⊥′X φ(ξ)

andφ(hX) = h′X

for every X ∈ TpM , ξ ∈ T⊥Mp and p ∈M .


Then there exists an isometry τ of M(c1)×M(c2) such that τ ◦ ψ = ψ′

and τ∗|T⊥M = φ.

Theorem 2.3.4. Let (M, g) be a simply connected n-dimensionalRiemannian manifold, ν a Riemannian vector bundle over M of rank mwith metric g, ∇⊥ a connection on ν compatible with the metric g, σ asymmetric (1, 2) tensor with values in ν and let f : TM → TM , t : ν → νand h : TM → ν be (1, 1) tensors over M . Define S : ν → End(TM)by g(SξX,Y ) = g(σ(X,Y ), ξ) and define s : ν → TM by g(sξ,X) =g(ξ, hX) for X,Y ∈ TM , ξ ∈ ν. Suppose that (M, g, ν, g, σ,∇⊥, f, h, t)satisfies the compatibility equations for M(c1) ×M(c2). Then, thereexists an isometric immersion ψ from M into M(c1)×M(c2) such thatσ and S are the second fundamental form and shape operator of M inM(c1) ×M(c2), respectively, ν is isomorphic to the normal bundle ofψ(M) in M(c1)×M(c2) by an isomorphism ψ : ν → T⊥ψ(M) and suchthat

F (ψ∗X) = ψ∗(fX) + ψ(hX), (2.46)

andF (ψξ) = ψ∗(sξ) + ψ(tξ), (2.47)

where F is the product structure of M(c1)×M(c2).

In the next section we will proof Theorems 2.3.3 and 2.3.4. We will proofTheorem 2.3.3 using techniques of [18] and Theorem 2.3.4 will be provenusing techniques of [24].

For ease of reference, we want to have an explicit statement of Theorems2.3.3 and 2.3.4 in the case of hypersurfaces. We omit the proof, becausethe theorem can be analogously proven as Theorems 2.3.3 and 2.3.4.

Theorem 2.3.5. Let (M, g) be a simply connected Riemannian manifoldof dimension n. Let S and f be symmetric fields of operators on M ,U a vector field on M and λ a smooth function on M . Assume that(g, S, f, U, λ) satisfies the compatibility equations for M(c1)×M(c2):

g(fX, Y ) = g(X, fY ) f2X = X − g(U,X)U fU = −λU,

g(U,U) + λ2 = 1, (2.48)

R(X,Y )Z = (SX ∧ SY )Z + c1

(X + fX

2 ∧ Y + fY

2

)Z


+c2

(X − fX

2 ∧ Y − fY2

)Z, (2.49)

(∇Y S)(X)− (∇XS)(Y ) =(c1I + f

4 − c2I − f

4

)(X ∧ Y )U, (2.50)

(∇Xf)(Y ) = u(Y )SX + g(SX, Y )U, (2.51)

∇XU = λSX − fSX, (2.52)

X[λ] = −2u(SX). (2.53)

Then there exists an isometric immersion ψ : M →M(c1)×M(c2) suchthat the shape operator with respect to the normal ξ associated to ψ is Sand such that

F (ψ∗X) = ψ∗(fX) + g(U,X)ξ,

F (ξ) = ψ∗(U) + λξ,

for every tangent vector field X on M and in which F is the productstructure of M(c1)×M(c2). Moreover the immersion is unique up to aglobal isometries of M(c1)×M(c2).Remark 2.3.6. We give here the relation between the formalism of Danieland our formalism. By definition of the product structure F and thedefinition of the vector field T and the function ν, we obtain

I − F2 X = g(X,T )T + g(X,T )νN,

I − F2 N = νT + ν2N,

and hence we haveFX = X − 2g(X,T )T − 2νg(X,T )N,

FN = −2νT + (1− 2ν2)N.Using the definition of f, u, U and λ and comparing the tangent andnormal components, we can conclude that

fX = X − 2g(X,T )T,

u(X) = −2νg(X,T ),

U = −2νT,

PROOFS OF THE MAIN THEOREMS 21

λ = (1− 2ν2).

From these equations and equations (2.3) and (2.4), one can deduceequations (2.51), (2.52) and (2.53).

2.4 Proofs of the main theorems

Proof of Theorem 2.3.3. We give the proof in the case c1 > 0 and c2 > 0.The other cases can be proven analogously. Consider the map C : M →Gl(R, n+m+ 2), defined by

Cp(ψ∗X) = ψ′∗X,

Cp(ξ) = φ(ξ),

Cp(ξ) = ξ′,

Cp(ξ) = ξ′,

for all X ∈ TpM and ξ ∈ T⊥Mp. ξ and ξ are normal vector fields ofψ(M) in En+m+2 such that

ξ = (ψ1, . . . , ψn1+1, 0, . . . , 0)

andξ = (0, . . . , 0, ψ1, . . . , ψn2+1).

ξ′ and ξ′ are defined analogously for ψ′(M) in En+m+2. Let us denote theLevi-Civita connection of the Euclidean space En+m+2 by D. Remarkthat

DX ξ = 12(ψ∗X + F (ψ∗X)) (2.54)

andDXξ = 1

2(ψ∗X − F (ψ∗X)), (2.55)

where F is the product structure of M(c1)×M(c2). We show that C is aconstant map, by showing that DC = 0, i.e. DX(C(V ))− C(DXV ) = 0,for every vector field V along ψ and X tangent vector field of M .Supposefirst that V is tangent, i.e. V = ψ∗Y for some tangent vector field Y .


Then

DXC(ψ∗Y )− C(DXψ∗Y ) = DX(ψ′∗(Y ))− C(ψ∗(∇XY ) + σ(X,Y ))

+ 12c1g(X + fX, Y )ξ′ + 1

2c2g(X − fX, Y )ξ′

= ψ′∗(∇XY ) + σ′(X,Y )

− 12c1g(X + f ′X,Y )ξ′ − 1

2c2g(X − f ′X,Y )ξ′

− C(ψ∗(∇XY ) + σ(X,Y ))

+ 12c1g(X + fX, Y )ξ′ + 1

2c2g(X − fX, Y )ξ′

= 0.

Next we can assume that V is normal, i.e. V = ξ for some normal vectorfield of ψ(M) in M(c1)×M(c2). Then

DX (C(ξ))− C(DXξ) = −S′φξX +∇⊥′X φ(ξ)

− 12c1g(φξ, h′X)ξ′ + 1

2c2g(φξ, h′X)ξ′ − C(−SξX

+∇⊥Xξ −12c1g(ξ, hX)ξ + 1

2c2g(ξ, hX)ξ)

= 0.

Analogously one can prove that (DXC)(ξ1) = 0 and (DXC)(ξ2) = 0.This means that C is the same linear map for every p ∈M . Hence wecan identify the map C with τ ∈ Gl(n+m+ 2,R). Moreover it is easy tosee that τ ∈ O(n+m+ 2), that τ(ξ1) = ξ′1 and that τ(ξ2) = ξ′2. We cannow easy deduce from the previous that τ is the map we were lookingfor.

Proof of Theorem 2.3.4. We give only the proof in the case c1 > 0 andc2 = 0. The other cases can be proven analogously. Let TM be thetangent bundle of M and ν the given Riemannian bundle over M of rankm with metric g. Denote by N the trivial line bundle over M equippedwith the Euclidean metric. Let B = TM

⊕ν⊕N be the orthogonal


Whitney sum of Riemannian vector bundles with Riemannian metric g.Let ξ1 be a section of N such that g(ξ1, ξ1) = 1

c1. We define a connection

D on B by

DXY = ∇XY + σ(X,Y )− 12c1g(X + fX, Y )ξ1,

DXξ = −SξX +∇⊥Xξ −12c1g(ξ, hX)ξ1,

DXξ1 = 12(X + fX + hX),

for all vector fields X,Y and sections ξ of ν. It is easy verified that theconnection D is compatible with the metric g of B, i.e.

Xg(η1, η2) = g(DXη1, η2) + g(η1, DXη2)

for every X ∈ TM and every η1, η2 ∈ B. The curvature tensor ofB with connection D will be denoted by RD. The curvature tensorRD : TM × TM ×B → B is a trilinear map over the module C∞(M) ofsmooth functions on M defined by

RD(X,Y )η = DXDY η −DYDXη −D[X,Y ]η.

We will show that the connection D on B is flat, i.e. RD = 0.

Lemma 2.4.1. RD = 0.

Proof. We will only calculate RD(X,Y )Z for arbitrary vector fieldsX,Y, Z on M . We will obtain that RD(X,Y )Z = 0, because of theequations (2.40), (2.41) and (2.43). Using the definition of the connectionD, we have

DXDY Z = DX(∇Y Z + σ(Y, Z)− c1g(Y + fY

2 , Z)ξ)

= ∇X∇Y Z + σ(X,∇Y Z)− c1g(X + fX

2 ,∇Y Z)ξ

− Sσ(Y,Z)X +∇⊥Xσ(Y,Z)− c1g(hX2 , σ(Y,Z))ξ

− c1

(g(∇X(Y + fY

2 ), Z) + g(Y + fY

2 ,∇XZ))ξ

− c1g(Y + fY

2 , Z)(X + fX

2 + hX

2

)


and a similar equation when X and Y are interchanged. We also have

D[X,Y ]Z = ∇[X,Y ]Z + σ([X,Y ], Z)− c1g(I + f

2 [X,Y ], Z))ξ.

Hence we obtain that RD(X,Y )Z is given by

R(X,Y )Z + Sσ(X,Z)Y − Sσ(Y,Z)X − c1

(X + fX

2 ∧ Y + fY

2

)

+ (∇σ)(X,Y, Z)− (∇σ)(Y,X,Z)− c1h

((X ∧ Y )Z + fZ

2

)

− c1

(g((∇X(I + f

2 ))Y,Z)− g(

(∇Y (I + f

2 ))X,Z)

+ g(hX2 , σ(Y, Z))− g(hY2 , σ(X,Z)))ξ.

From equation (2.40), (2.41) and (2.43) we can conclude that RD(X,Y )Zvanishes for all tangent vector fields X,Y and Z. The cases RD(X,Y )ξand RD(X,Y )ξ1, where ξ is an arbitrary section of ν, can be treatedanalogously using equations (2.41), (2.42), (2.43) and (2.44). Since RDis a trilinear map, we obtain that RD(X,Y )η = 0 for every X,Y ∈ TMand every η ∈ B.

Define now a bundle map F : B → B by

FX = fX + hX,

Fξ = sξ + tξ,

Fξ1 = ξ1,

for all vector fields X,Y and sections ξ of ν. In the following two lemmaswe will show that F 2 = I, F is symmetric with respect to g and thecovariant derivative of the bundle map F on B is 0.

Lemma 2.4.2. F 2η = η and g(Fη1, η2) = g(η1, Fη2).

Proof. This follows immediately form the definition of the bundle mapF and equations (2.37), (2.38) and (2.39).


Lemma 2.4.3. (DXF )(η) = 0 for every X ∈ TM and every η ∈ B.

Proof. This follows immediately form the definition of the bundle map F ,the definition of connection D and equations (2.43), (2.44) and (2.45).

Let B1 and B−1 be subsets of B defined respectively by

{η ∈ B |Fη = η}

and{η ∈ B |Fη = −η}.

In the next proposition we will show that there exist orthonormal parallelsections η1, . . . , ηn1+1, η1, . . . , ηn2 such that F ηi = ηi and Fηα = −ηαfor i ∈ {1, . . . , n1 + 1} and α ∈ {1, . . . , n2}.

Proposition 2.4.4. Let (M, g) be a Riemannian manifold that satisfiesthe assumptions of Definition 1. Suppose that c1 > 0 and c2 = 0. LetB be the vector bundle over M as defined above and F the bundle mapof B as defined above, then there exist orthonormal parallel sectionsη1, . . . , ηn1+1, η1, . . . , ηn2 such that η1, . . . , ηn1+1 ∈ B1 and η1, . . . , ηn2 ∈B−1.

Proof. Using the fact that I+F2 and I−F

2 are of rank n1 + 1 and n2,respectively, and the fact that F is symmetric, we obtain an orthonormalbasis v1, . . . , vn1+1, v1, . . . , vn2 of Bp such that F vi = vi and Fvα =−vα for i ∈ {1, . . . , n1 + 1} and α ∈ {1, . . . , n2}. Since D is a flatconnection on B and M is simply connected, there exist parallel sectionsη1, . . . , ηn1+1, η1, . . . , ηn2 such that ηi(p) = vi and ηα(p) = vα. Moreover,since the connection D is compatible with the metric g, we have thatη1, . . . , ηn1+1, η1, . . . , ηn2 are parallel orthonormal sections on B. Fromthe fact that DXF = 0 for every tangent vector field X, we obtain thatF ηi = ηi and Fηα = −ηα.

We are ready to construct an isometric immersion ψ ofM in Sn1(c1)×En2 ,with n1+n2 = n+m. Denote by ωα the metric dual 1-form of ηα. It is easyverified by direct calculations that dωα = 0, i.e. ωα is closed. Since M issimply connected, there exist functions ψα such that dψα = ωα. Denoteby ψi the functions g(ηi, ξ1). Define a mapping ψ : M → En+m+1 byψ = (ψ1, . . . , ψn1+1, ψ1, . . . , ψn2), where En+m+1 denotes the Euclideanspace. It is easy verified that ψ(M) ⊂ Sn1(c1)× En2 . We show by direct


calculations that ψ is an isometric immersion. Let v ∈ TpM be a tangentvector of M at a point p such that ψ∗(v) = 0, then by definition of ψ wehave that

g(ηi,12(v + fv + hv)) = 0

andg(v, ηα) = 0

for every i and α. Hence v = 0, because Fv = v if g(v, ηα) = 0. Since vand p were arbitrary, we obtain that ψ is an immersion. We also havethat

〈ψ∗(v), ψ∗(w)〉 = 14

n1+1∑i=1

g(v + Fv, ηi)g(w + Fw, ηi)

+n2∑α=1

g(v, ηα)g(w, ηα)

=n1+1∑i=1

g(v, ηi)g(w, ηi) +n2∑α=1

g(v, ηα)g(w, ηα) = g(v, w)

for any v, w ∈ TpM and p ∈ M . We conclude that ψ is an isometricimmersion into Sn1(c1)× En2 . We define now a isomorphism Ψ betweenthe bundle B and TRn+m+1 restricted to ψ(M) by Ψ(ηi) = ei andψ(ηα) = en1+1+α, where eγ , γ ∈ {1, . . . , n+m+ 1}, is the restriction ofthe canonical frame of TRn+m+1 to ψ(M). For a tangent vector field Xof M we have

Ψ(X) =n1+1∑i=1

g(X, ηi)ei +n2∑α=1

g(X, ηα)en1+1+α

=n1+1∑i=1

g(12(X + FX), ηi)ei +

n2∑α=1

g(X, ηα)en1+1+α

= ψ∗(X)

and for ξ1 we have

Ψ(ξ1) =n1+1∑i=1

g(ξ1, ηi)ei.

MINIMAL SURFACES IN M(C1)×M(C2) 27

Hence Ψ sends the tangent bundle TM to the tangent bundle of ψ(M)and ξ1 to the first n1 + 1 components of ψ. Since Ψ is an isometry in thefibres, it sends the Riemannian bundle ν to the normal bundle of ψ(M)in Sn1(c1)×En2 . Define by ψ the restriction of Ψ to ν. ψ is obviously anisomorphism between the bundles ν and T⊥ψ(M). Moreover, we havethat Ψ sends parallel orthonormal sections η1, . . . , ηn1+1, η1, . . . , ηn2 intothe parallel orthonormal sections e1, . . . , en+m+1 of TRn+m+1 and hence

Ψ(DXY ) = ∇ψ∗(X)ψ∗(Y ) (2.56)

Ψ(DXξ) = ∇ψ∗(X)ψ(ξ) (2.57)

andΨ(DXξ1) = ∇ψ∗(X)Ψ(ξ1), (2.58)

where ∇ denotes the Levi-Civita connection of TRn+m+1. Define abundle map F of TEn+m1|ψ(M) by FΨ = ΨF . It is easy to see thatF ei = ei and that F eα = −eα. Hence we obtain that F 2 = I andthat F is symmetric. Moreover, one can see that F is parallel alongψ(M). It is easy to see that F is a restriction of the product structure ofSn1 × En2 to ψ(M). Comparing the tangent and the normal componentsof (2.56), (2.57) and (2.58) and using the fact that FΨ = ΨF , we obtainthat σ is the second fundamental form of ψ(M) in Sn1(c)×En2 and thatthe equations (2.46) and (2.47) hold.

2.5 Minimal surfaces in M(c1)×M(c2)

In this section we will proof the existence of a family of minimal surfacesin M(c1)×M(c2). This theorem is a generalization of Theorem 4.2 in[14] of Daniel. We will first proof a lemma and a proposition to proof themain theorem of this section. Let us fix the notations for this section.

Let Σ be a Riemann surface with metric g, ∇ its Levi-Civita connectionand J the rotation of angle π

2 on TΣ. We have then

J2 = −I, g(JX, Y ) = −g(X, JY ) and (∇XJ)Y = ∇XJY − J∇XY = 0.

We call J the complex structure of Σ. Let ν be a Riemannian bundle ofrank m over Σ with metric g, ∇⊥ a connection on ν compatible with g,


σ a (1, 2) tensor field over Σ with values in ν. Let f, h, s and t be tensorfields over Σ as defined in definition 2.3.2 and define S : ν → End(TΣ)by g(SξX,Y ) = g(σ(X,Y ), ξ) for very X,Y ∈ TΣ and ξ ∈ ν.

Lemma 2.5.1. Let Σ be a Riemann surface, J the complex structure ofΣ and A a (1, 1) tensor field over Σ, then

AJX ∧AY = −AX ∧AJY, (2.59)

AJX ∧AJY = AX ∧AY, (2.60)

J(X ∧ Y ) = (X ∧ Y )J. (2.61)

Proof. By direct computations one can show that the equations of thelemma hold.

Proposition 2.5.2. Assume that Sξ is trace-free for every ξ ∈ ν and(Σ, g, ν, g, σ,∇⊥, f, h, t) satisfies the compatibility equations for M(c1)×M(c2). For θ ∈ R we set

Sθξ = TθSξ, for every ξ ∈ ν, (2.62)

fθ = TθfT−θ, (2.63)

hθ = hT−θ, (2.64)

tθ = t, (2.65)

where Tθ = cos(θ)I + sin(θ)J . Then Sθξ is trace-free and symmetric forevery ξ ∈ ν and (Σ, g, ν, g, σθ,∇⊥, fθ, hθ, tθ) satisfies the compatibilityequations for M(c1)×M(c2).

Proof. We define first sθ by Tθs, where s is the transpose of h withrespect to g. The equations g(SξX,Y ) = g(X,SξY ) and trSξ = 0 implythat Sθξ is trace-free and symmetric for every ξ. It is easy to see that fθand tθ are symmetric, f2

θ = I − sθhθ and t2θ = I − hθsθ. By definition ofsθ, we see that sθ is the the transpose of hθ. We show that fθ, hθ and tθsatisfy equations (2.43), (2.44) and (2.45).

(∇Xfθ) = (∇X(TθfT−θ))Y

= Tθ(∇Xf)T−θY


= Tθ(ShT−θYX + sσ(X,T−θY )

)= SθhθYX + sθσθ(X,Y ),

∇⊥XhθY − hθ∇XY = ∇⊥XhT−θY − hT−θ∇XY

= ∇⊥Xh(T−θY )− h∇X(T−θY )

= tσ(X,T−θY )− σ(X, fT−θY )

= tσ(TθX,Y )− σ(TθX,TθfT−θY )

= tθσθ(X,Y )− σθ(X, fθY )

and

∇⊥Xtθξ − tθ∇⊥Xξ = ∇⊥Xtξ − t∇⊥Xξ

= −σ(X, sξ)− hSξX

= −σ(TθX,Tθsξ)− (hT−θ)(TθSξX)

= −σθ(sθξ,X)− hθSθξX.

We show now that equations of Gauss, Codazzi and Ricci are satisfied.We would like to remark that the equation of Codazzi (2.41) is equivalentto

(∇Y Sξ)X − (∇XSξ)Y =(c1

(I + f

2

)− c2

(I − f

2

))(X ∧ Y )sξ2 ,

where (∇Y Sξ)X = ∇Y SξX − Sξ∇YX − S∇⊥XξY . We notice also, as

dimΣ = 2, that the equation of Gauss is equivalent to

K =m∑α=1

det(Sξα) + c1det(I + f

2

)+ c2det

(I − f

2

),

where {ξ1, . . . , ξm} is an orthonormal frame of T⊥Σ and K the Gaussiancurvature of Σ and hence we see that the equation of Gauss is satisfiedfor (Σ, g, ν, g, σθ,∇⊥, fθ, hθ, tθ), because det(Tθ) = 1, det

(I+fθ

2

)=

det(Tθ(I+f

2

)T−θ

)= det

(I+f

2

)and det

(I−fθ

2

)= det

(I−f

2

). To prove


the equation of Codazzi, we remark first that

(∇XSθξ )Y = ∇XSθξY − Sθξ∇XY − Sθ∇⊥XξY

= Tθ(∇XSξY − Sξ∇XY − S∇⊥XξY

)= Tθ

(∇XSξ

)Y,

and hence we have, using lemma 2.5.1,

(∇Y Sθξ )X − (∇XSθξ )Y = Tθ((∇Y Sξ)X − (∇XSξ)Y

)= Tθ

(c1I + f

2 − c2I − f

2

)(X ∧ Y )sξ2

=(c1Tθ

I + f

2 − c2TθI − f

2

)(T−θTθ) (X ∧ Y )sξ2

=(c1I + fθ

2 − c2I − fθ

2

)Tθ(X ∧ Y )sξ2

=(c1I + fθ

2 − c2I − fθ

2

)(X ∧ Y )Tθ

sξ

2

=(c1I + fθ

2 − c2I − fθ

2

)(X ∧ Y )sθξ2 .

Finally we proof that the Ricci equation (2.42) holds. It is sufficient toshow that

c1 + c24 (hθX ∧ hθY ) ξ + σθ(SθξY,X)− σθ(SθξX,Y )

= c1 + c24 (hX ∧ hY ) ξ + σ(SξY,X)− σ(SξX,Y ).

This equation follows directly from lemma 2.5.1. We can concludethat (Σ, g, ν, g, σθ,∇⊥, fθ, hθ, tθ) satisfies the compatibility equations forsurfaces in M(c1)×M(c2).

Theorem 2.5.3. Let Σ be a simply connected Riemann surface andx : Σ→ M(c1)×M(c2) a conformal minimal immersion. Let T⊥Σ bethe normal bundle of Σ, σ the second fundamental form and ∇⊥ the


normal connection of Σ inM(c1)×M(c2). Let f, h, s and t be (1, 1) tensorfields over Σ defined by (2.9) and (2.10). Let p0 ∈ Σ. Then there exists aunique family of minimal conformal immersions xθ : Σ→M(c1)×M(c2)such that:

1. xθ(p0) = x(p0) and (dxθ)p0 = (dx)p0,

2. the metrics induced by xθ and x are the same,

3. the second fundamental form of xθ(Σ) in M(c1)×M(c2) is givenby σθ(X,Y ) = σ(TθX,Y ) for every X,Y ∈ TΣ,

4. F (dxθX) = dxθ(fθX) + hθX and Fξ = dxθsθξ + tθξ for everyX ∈ TΣ and ξ ∈ T⊥Σ.

Moreover we have x0 = x and the family xθ is continuous with respectto θ. The family of immersions (xθ)θ∈R is called the associate familyof immersion x : Σ → M(c1) ×M(c2). The immersion xπ

2is called

the conjugate immersion of x. The immersion xπ is called the oppositeimmersion of x.

Proof. Let g be the metric of Σ induced by x. We have nowthat (Σ, g, T⊥Σ, g, σ,∇⊥, f, h, t) satisfies the compatibility equationsof M(c1) × M(c2). Hence by previous proposition we obtain that(Σ, g, T⊥Σ g, σθ,∇⊥, fθ, hθ, tθ) satisfies the compatibility equations forM(c1)×M(c2). Thus by theorems 2.3.3 and 2.3.4 there exist a uniqueimmersion xθ satisfying the properties of the theorem. The fact thatx0 = x is clear. By construction of the immersion we see that the family(xθ)θ∈R is continuous with respect to θ.

Chapter 3

Constant angle surfaces inproduct spaces

3.1 Introduction

In recent years a lot of people started the study of submanifolds inproduct spaces, in particular surfaces M2 in M2(c)× R, where M2(c) isa 2-dimensional space form of curvature c 6= 0. This was initiated bythe study of minimal surfaces in the product space M2 × R by Meeksand Rosenberg in [43] and by Rosenberg in [47]. In the papers [21] and[22] geometers began the study of constant angle surfaces in M2(c)× R,i.e. surfaces for which the normal of the surface makes a constant anglewith the vector field ∂t parallel to the second component of M2(c)× Rand hence also with the first component TpM2(c) of Tp(M2(c) × R).They proved that they can construct all the constant angle surfaces inM2(c) × R starting from an arbitrary curve in M2(c) and that thesesurfaces have constant Gaussian curvature.Theorem 3.1.1 ([21] and [22]). A surface M2 immersed in M2(c)×R isa constant angle surface if and only if the immersion ψ is (up to isometriesof M2(c)× R) locally given by ψ : M2 → M2(c)× R : (u, v) 7→ ψ(u, v),where ψ(u, v) is given by

(cos(√c cos(θ)u)f(v) + sin(

√c cos(θ)u)f(v)× f ′(v), sin(θ)u)

if c > 0, or by

(cosh(√−c cos(θ)u)f(v) + sinh(

√−c cos(θ)u)f(v) � f ′(v), sin(θ)u)

32

CURVES IN M2(C). 33

if c < 0, in which f : I → M2(c) is a unit speed curve in M2(c) andθ ∈ [0, π] is the constant angle.

Here we would like to define and classify constant angle surfaces in aproduct space M2(c1)×M2(c2) of two 2-dimensional space forms, notboth flat. We show that these constant angle surfaces have necessarilyconstant Gaussian curvature. In the classification theorem we show thatsome of the constant angle surfaces can be constructed from curves inM2(c1) and M2(c2). But in some cases, the constant angle surfaces inM2(c1)×M2(c2) will be constructed from a solution of a Sine-Gordonequation and its Bäcklund transformation.

This chapter is organized as follows. In the first section we will recall somedefinitions and notations about curves in two dimensional space formsM2(c). In section 3.3 we define constant angle surfaces inM2(c1)×M2(c2)and show that these surfaces have constant Gaussian curvature andconstant normal curvature. We also show that totally geodesic surfacesin M2(c1) ×M2(c2) have necessarily constant angles. We choose alsoa canonical frame in order to simplify our calculations in the followingsections. In section 3.4 we state some existence results in order to givea classification of the constant angle surfaces. These existence resultswill have an connection with the Sine-Gordon and Sinh-Gordon equationtogether with their Bäcklund transformation. We finish this chapter withthe classification of all the constant angle surfaces. One can find theresults of this chapter also in [25].

3.2 Curves in M 2(c).

In this short section we will discus curves in 2-dimensional space formsM2(c) with c 6= 0. It is known that M2(c) is isometric to the 2-dimensional sphere S2(c) of radius 1√

cif c > 0, i.e.

S2(c) = {(p1, p2, p3) ∈ E3 | p21 + p2

2 + p23 = 1

c},

endowed with the induced metric of E3. The tangent space TpS2(c) inevery point p is given by

S2(c) = {v ∈ TpE3 | 〈p, v〉 = 0}.

34 CONSTANT ANGLE SURFACES IN PRODUCT SPACES

Using the cross-product × in E3, we define a complex structure J onTS2(c) by

J : TS2(c)→ TS2(c) : vp 7→√c(p× v)p.

It is easy to see that if v ∈ TpS2(c) and ‖v‖2 = 1, then {v, Jv} is anorthonormal basis of TpS2(c).

We can define in a similar manner a complex structure when c < 0. Itis known that M2(c) is isometric to the hyperbolic plane H2(c) if c < 0.We use here the Minkowski or the hyperboloid model of the hyperbolicplane. Denote by R3

1 the Minkowski 3-space with standard coordinatesp1, p2 and p3, endowed with the Lorentzian metric

〈., .〉1 = −dp21 + dp2

2 + dp23.

The hyperbolic plane H2(c) can be constructed as the upper sheet (p1 > 0)of the hyperboloid

{(p1, p2, p3) ∈ R31 | − p2

1 + p22 + p2

3 = 1c},

endowed with the induced metric of R31. The tangent space TpH2(c) in

every point p is given by

TpH2(c) = {v ∈ TpR31 | 〈p, v〉1 = 0}.

Using the Lorentzian cross-product � in R31(see for example [22]), we

define a complex structure J on TH2(c) by

J : TH2(c)→ TH2(c) : vp 7→√−c(p� v)p.

It is easy to see that if v ∈ TpH2(c) and ‖v‖2 = 1, then {v, Jv} is anorthonormal basis of TpH2(c). In the following we will denote J as thecomplex structure of M2(c).

Let α : I → M2(c) be an arclength parameterized curve in M2(c).Denote by T (s) ∈ Tα(s)M

2(c) the tangent unit vector α′(s) and byN(s) ∈ Tα(s)M

2(c) the normal vector JT (s). By direct calculations, onecan show that

T ′ = DTT = κN − cα,N ′ = DTN = −κT,

where D is the Levi-Civita connection of E3 or of R31. We call κ the

geodesic curvature of α in M2(c). We will need the geodesic curvatureof a curve in M2(c) in order to state our classification results of constantangle surfaces.

CONSTANT ANGLE SURFACES 35

3.3 Constant angle surfaces

Let M2 be a surface of M2(c1) ×M2(c2) and let f, h, s and t be (1, 1)tensor fields over M2 as defined in the previous chapter. Since f is asymmetric (1, 1) tensor on M2, there exist continuous functions λ1 ≤ λ2on M2 such that for every p in M2 λ1(p) and λ2(p) are eigenvaluesof f at p. Moreover λ1 and λ2 are differentiable functions in pointswhere λ1 and λ2 are different. Assume that λ1 < λ2, then one can showthat the distributions Tλ1 = {X ∈ TM2|fX = λ1X} and Tλ2 = {X ∈TM2|fX = λ2X} are differentiable. From the fact that F 2 = I andg(FX,FY ) = g(X,Y ), is it easy to deduce that λ2

i ≤ 1 for i = 1, 2.Hence we have that for every point p there exist unique θ1(p) and θ2(p)in [0, π2 ] such that

λ1(p) = cos(2θ1(p)) and λ2(p) = cos(2θ2(p)).

We call θ1 and θ2 the angle functions of M2 in M(c1) ×M(c2). Thisdefinition is inspired by the definition of angles between 2-dimensionallinear subspaces of the Euclidean space E4 given in [60], where θ1(p) andθ2(p) are the angles between TpM2 and Tp(M2(c1)×{p2}), p = (p1, p2) ∈M . Moreover this definition of angle for surfaces in M2(c1) ×M2(c2)coincides with definition of angle for surfaces in M2(c)× R.For Lagrangian surfaces in S2 × S2, a similar notion for angle wasintroduced in [35]; since for Lagrangian surfaces λ1 + λ2 = 0, see below,there is only one angle function. Lagrangian surfaces in S2 × S2 are alsostudied in [6].

Definition 3.3.1. A surface in M2(c1) ×M2(c2) is a constant anglesurface if θ1 and θ2 are constant.

This definition also makes sense for c1 = c2 = 0, but in this case it isbetter to call a surface in E4 a constant angle surface if there is a fixedplane in E4 such that TpM makes constant angles with this plane.

3.3.1 Complex structures

Let J and J be complex structures on M2(c1)×M2(c2) defined by

Jv = J(v1, v2) = (J1v1, J2v2) = (J1I + F

2 v, J2I − F

2 v)


and

Jv = J(v1, v2) = (J1v1,−J2v2) = (J1I + F

2 v,−J2I − F

2 v),

respectively, where J1 and J2 denote the standard complex structureson M2(c1) and M2(c2). We obtain the following connection between theangle functions and the complex structures J and J .

Proposition 3.3.2. Consider a surface M2 in M2(c1) ×M2(c2) withangle functions θ1 and θ2, then

g(Jv, w) = cos(θ1 − θ2)ωM2(v, w) or cos(θ1 + θ2)ωM2(v, w) (3.1)

and

g(Jv,w) = cos(θ1 + θ2)ωM2(v, w) or cos(θ1 − θ2)ωM2(v, w), (3.2)

for all v, w ∈ TpM2 and p ∈ M and a suitable choice of volume formωM2 of M2.

Proof. We only prove this proposition in the case that c1, c2 > 0. Theother cases can be proved analogously. Let us consider an orthonormalbasis {e1, e2} of TpM2 that diagonalizes f . Hence we have that fei =cos(2θi)ei for i = 1, 2. Then

g(Je1, e2) = −√c1(I + F

2 e1 ×I + F

2 e2) · p1

−√c2(I − F2 e1 ×

I − F2 e2) · p2

= ε1(‖I + F

2 e1‖‖I + F

2 e2‖+ ε2‖I − F

2 e1‖‖I − F

2 e2‖)

= ε1 cos(θ1 − θ2) or ε1 cos(θ1 + θ2),

where ε21 = ε22 = 1. This proves the first equation in the proposition. Thesecond equation can be proved similarly and the proof of the propositionis finished.

By direct computations, one can now easily prove the following theorem.

Theorem 3.3.3. A surface M in M2(c1)×M2(c2) is a complex surfacewith respect to J or J if and only if f is proportional to the identity.M2 is Lagrangian with respect to J or J if and only if the trace of fvanishes.


3.3.2 Totally geodesic surfaces

We show now that totally geodesic surfaces in M2(c1) × M2(c2) areconstant angle surfaces in M2(c1)×M2(c2).

Proposition 3.3.4. SupposeM2 is a totally geodesic surface ofM2(c1)×M2(c2), then M2 is a constant angle surfaces in M2(c1)×M2(c2).

Proof. As M2 is a totally geodesic surface, we have that (∇Xf) = 0 forany X ∈ TM2 and hence the eigenvalues of f are constant. Let p be anarbitrary point in M2(c1)×M2(c2) and {e1, e2} an orthonormal basis inTpM

2 such that fe1 = λ1e1 and fe2 = λ2e2. Using equation of Codazzi,we obtain

0 = (aλ1 + b)(1− λ22) = (aλ2 + b)(1− λ2

1),

with a = c1+c24 and b = c1−c2

4 . Hence we obtain that λ1 = λ2 = − ba with

c1c2 > 0, λ21 = λ2

2 = 1, λ21 = 1 and λ2 = − b

a with c1c2 > 0 or λ21 = 1 and

λ2 = cos(2θ) with c1c2 = 0 and θ ∈ [0, π2 ]. This conditions are equivalentto

1. λ1 = λ2 = − ba with c1c2 > 0,

2. λ1 = ±1 and λ2 = ±1,

3. λ1 = ±1 and λ2 = − ba with c1c2 > 0, and

4. λ1 = 1 and λ2 ∈ [−1, 1] with c1 = 0 or λ1 = −1 and λ2 ∈ [−1, 1]with c2 = 0.

Since λ1 and λ2 are continuous, one of the above conditions must hold.Hence we obtain that totally geodesic surfaces of M2(c1)×M2(c2) areconstant angle surfaces, in which λ1 and λ2 have one of the above specificvalues.

In this paragraph we will give a local classification of totally geodesicsurfaces for which λ1 = λ2 = − b

a with c1c2 > 0. The other cases will betreated in the next sections and will appear as special cases of constantangle surface. We classify the totally geodesic surfaces ofM2(c1)×M2(c2)in Theorem 3.5.4. Suppose that c1, c2 > 0, the other case can be treatedanalogously and the result of the second case is stated together with thefirst case in Proposition 3.3.5.


We can immerse M2(c1)×M2(c2) as a submanifold of codimension 2 inthe Euclidean space E6. We also remark, by using the equation of Gauss,that we obtain that the surface M2 has constant Gaussian curvaturec1c2c1+c2

. So let ψ : M2 →M2(c1)×M2(c2) be a totally geodesic surface inM2(c1)×M2(c2) with λ1 = λ2 = − b

a . Let us fix a point p in an open setU of M2 and let (u, v) be Fermi coordinates of U in M2, there alwaysexist such coordinates on an open set of a surface (see for example [38]).The metric g of M has then the form

du2 +G(u, v)dv2

on the open set U of M , with G(0, v) = 1 and ∂∂uG(0, v) = 0 for every

v, in terms of the Fermi coordinates (u, v). Since M2 has constantGaussian curvature, we have that G is uniquely determined by thepartial differential equation

∂2

∂2u

√G = −K

√G,

where K is the Gaussian curvature of M2. So we find that G is given by

cos2(√Ku), (3.3)

because of the initial conditions G(0, v) = 1 and ∂∂uG(0, v) = 0 for every

v. Let us now consider M2 as a surface of codimension 4 immersed inE6. The formulas of Gauss are then given by

D∂u∂u = −K−→x , (3.4)

D∂u∂v = D∂v∂u = −√K tan(

√Ku)∂v, (3.5)

D∂v∂v =√K cos(

√Ku) sin(

√Ku)∂u −K cos2(

√Ku)−→x , (3.6)

where −→x is the position vector of M2 in E6. Solving equations (3.4) and(3.5) we find that ψ is locally given by

(cos(√Ku)f1(v) + sin(

√Ku)g1, . . . , cos(

√Ku)f3(v) + sin(

√Ku)g3,

cos(√Ku)f1(v) + sin(

√Ku)g1, . . . , cos(

√Ku)f3(v) + sin(

√Ku)g3),


where g = (g1, g2, g3) and g = (g1, g2, g3) are constant vectors in R3.Moreover we have the following conditions

g(ψu, ψu) = 1, g(ψu, ψv) = 0, g(ψv, ψv) = cos2(√

c1c2c1 + c2

u),

g(ψu, h∂u) = 0, g(ψu, h∂v) = 0, , g(h∂u, h∂v) = 0,

g(ψv, h∂v) = 0, g(ψv, h∂u) = 0,

g(h∂u, h∂u) = 4c1c2(c1 + c2)2 g(h∂v, h∂v) = 4c1c2

(c1 + c2)2 cos2(√

c1c2c1 + c2

u),

g(ψu, ξ) = 0, g(ψv, ξ) = 0, g(ξ, ξ) = 1c1,

g(ψu, ξ) = 0, g(ψv, ξ) = 0, g(ξ, ξ) = 1c2,

g(h∂u, ξ) = g(h∂v, ξ) = g(h∂u, ξ) = g(h∂v, ξ) = 0,

in which ξ = (ψ1, ψ2, ψ3, 0, 0, 0) and ξ = (0, 0, 0, ψ4, ψ5, ψ6). Theseconditions are equivalent to

3∑i=1

f2i =

3∑i=1

g2i = 1

c1,

3∑j=1

f2j =

3∑j=1

g2j = 1

c2,

3∑i=1

figi =3∑i=1

f ′i gi = 0,

3∑j=1

f jgj =3∑j=1

f′jgj = 0,

3∑i=1

(f ′i)2 = c2c1 + c2

,3∑j=1

(f ′i)2 = c1c1 + c2

.

From the above equations we can conclude that f = (f1, f2, f3) andf = (f1, f2, f3) are curves inM2(c1) andM2(c2), respectively. Moreover


we see that f and f are curves of speed√

c2c1+c2

and√

c1c1+c2

, respectively.The constant vector g is perpendicular to the vectors f and f ′ and theconstant vector g is perpendicular to the vectors f and f ′. Hence weobtain that g = ±

√c1+c2c2

f × f ′ and g = ±√

c1+c2c1

f × f ′. Since g andg are constant vectors we obtain that the curves f and f are circles ofradius 1√

c1and 1√

c2, respectively. We obtain the following proposition.

Proposition 3.3.5. Let ψ : M2 →M2(c1)×M2(c2) be a totally geodesicsurface with λ1 = λ2 = − b

a , then ψ is locally congruent to

(cos(√

c1c2c1 + c2

u)f(v) + sin(√

c1c2c1 + c2

u)√c1 + c2c2

f(v)× f ′(v),

cos(√

c1c2c1 + c2

u)f(v) + sin(√

c1c2c1 + c2

u)√c1 + c2c1

f(v)× f ′(v)), (3.7)

where f and f are geodesic circles in M2(c1) and M2(c2), respectively,of constant speed

√c2

c1+c2and

√c1

c1+c2if c1, c2 > 0 or to

(cosh(√− c1c2c1 + c2

u)f(v) + sinh(√− c1c2c1 + c2

u)√c1 + c2c2

f(v) � f ′(v),

cosh(√− c1c2c1 + c2

u)f(v) + sinh(√− c1c2c1 + c2

u)√c1 + c2c1

f(v) � f′(v)),

(3.8)

where f and f are geodesic curves in M2(c1) and M2(c2), respectively,of constant speed

√c2

c1+c2and

√c1

c1+c2if c1, c2 < 0.

3.3.3 f is proportional to the identity

Suppose now that f = λI, and that λ = cos(2θ) is a constant. Usingequations (2.11) and (2.17) we see that g(hX, hY ) = sin2(2θ)g(X,Y )for every X,Y ∈ TM2. Moreover from equation (2.18) we immediatelydeduce that ShXY + s(σ(X,Y )) = 0. Suppose first that sin 2θ = 0 andhence we obtain that θ = 0 or θ = π

2 . In the first case this means that


the tangent vector fields along M2 are eigenvectors of F with eigenvalue1 and that the normal vector fields along M2 are eigenvectors of Fwith eigenvalue −1. It can be shown then that M2 is an open partof M2(c1) × {p2}. Analogously we obtain that M2 is an open part of{p1} ×M2(c2) if θ = π

2 .

Let θ be now a constant in (0, π2 ). Using the fact that ShXY +s(σ(X,Y )) = 0 for every X,Y ∈ TM2, we deduce that v is an eigenvectorof Shv with eigenvalue 0 for every v ∈ TpM2, i.e. Shvv = 0. Take nowan arbitrary orthonormal basis {e1, e2} ⊂ TpM

2. Consider the shapeoperators She1 and She2 associated to he1 and he2, respectively. We havethen that She1e2 = µ1e2 and She2e1 = µ2e1. Moreover we have that0 = Sh(e1+e2)(e1 +e2) = µ1e2 +µ2e1 and hence we have that µ1 = µ2 = 0.We conclude that M2 is a totally geodesic surface in M2(c1)×M2(c2),because {he1, he2} is an orthogonal basis of T⊥M2 and She1 = She2 = 0.Using the equation of Codazzi, we obtain that

(c1 cos2(θ)− c2 sin2(θ)) sin(θ) cos(θ) = 0.

Since θ ∈ (0, π2 ) and c1 and c2 are not both 0, we obtain that c1c2 > 0and tan2(θ) = c1

c2. Hence we have that cos(2θ) = c2−c1

c1+c2. Moreover we

have that the Gauss curvature of the surface equals c1c2c1+c2

. We summarizethe previous in the following proposition.

Proposition 3.3.6. Let M2 be a surface immersed in M2(c1)×M2(c2).Suppose f = λI with λ ∈ [−1, 1] and λ is a constant. Then M2 is atotally geodesic surface in M2(c1)×M2(c2). Moreover we have that λis equal to 1,−1 or c2−c1

c1+c2with c1c2 > 0 and that the Gauss curvature

of M2 equals c1, c2 and c1c2c1+c2

. Hence the surface is an open part ofM2(c1)× {p2}, {p1} ×M2(c2) or is locally given by (3.7) or (3.8).

3.3.4 f is not proportional to the identity

We first consider the trivial case θ2 = 0 and θ1 = π2 . One can then easily

prove that M2 is an open part of an Riemannian product of a curvein M2(c1) and a curve of M2(c2). One can now also easily check thatthis type of surface is totally geodesic if and only if both curves of theproduct are geodesic curves.

Suppose now that θ1 = π2 and θ2 is a constant in (0, π2 ). Denote in the

following θ2 by θ. Consider an adapted orthonormal frame {e1, e2, ξ1, ξ2}


such that fe1 = −e1, fe2 = cos(2θ)e2, tξ1 = ξ1 and tξ2 = − cos(2θ)ξ2.Using equations (2.11), (2.14) and (2.17), we see that he2 = ± sin(2θ)ξ2.We may suppose that he2 = sin(2θ)ξ2. Moreover we can deduce fromequations (2.11) and (2.17) that he1 = 0. Using equations (2.18), (2.19)and (2.20), we obtain that

2 cos2(θ)∇Xe2 = sin(2θ)Sξ2X + s(σ(X, e2)), (3.9)

− sin(2θ)g(∇Xe1, e2)ξ2 = t(σ(X, e1)) + σ(X, e1), (3.10)

2 cos2(θ)∇⊥Xξ2 = sin(2θ)σ(e2, X) + h(Sξ2X). (3.11)

From equations (3.9) and (3.10) we deduce that g(Sξ2X, e2) = g(Sξ1X, e1)= 0 for every X ∈ TM2. Hence we obtain, using equations (3.9) and(3.11) and the fact that g(Sξ2X, e2) = g(Sξ1X, e1) = 0, that

g(∇Xe1, e2) = − tan(θ)µ2g(X, e1),

g(∇Xξ1, ξ2) = − tan(θ)µ1g(X, e2),

where µ1 is the eigenvalue of Sξ1 and µ2 is the eigenvalue of Sξ2 . Sincewe know the shape operators Sξ1 and Sξ2 and the symmetric operatorf , we can find the Gaussian curvature K of M2. From the equationof Gauss we find that K = c2 sin2(θ). From the equation of Ricci weobtain also easily that K⊥ = |g(R⊥(e1, e2)ξ2, ξ1)| = 0. We summarizethe previous in the following proposition.Proposition 3.3.7. Let M2 be a constant angle surface immersed inM2(c1) × M2(c2). Suppose that λ1 = −1 and λ2 is a constant in(−1, 1). Then we can find an adapted frame {e1, e2, ξ1, ξ2} such thatfe1 = −e1, fe2 = cos(2θ)e2, tξ1 = ξ1 and tξ2 = − cos(2θ)ξ2, wherecos(2θ) = λ2 and such that the shape operators Sξ1 and Sξ2 take thefollowing form with respect to the orthonormal frame {e1, e2} :

Sξ1 =(

0 00 µ1

), Sξ2 =

(µ2 00 0

), (3.12)

for some functions µ1 and µ2 on M2. Moreover the Levi-Civitaconnection ∇ of M2 and the normal connection ∇⊥ of M2 in M2(c1)×M2(c2) are given by

g(∇Xe1, e2) = − tan(θ)µ2g(X, e1), (3.13)

g(∇⊥Xξ1, ξ2) = − tan(θ)µ1g(X, e2). (3.14)


The Gaussian curvature K is given by

K = c2 sin2(θ), (3.15)

and the normal curvature K⊥ is equal to 0.

We obtain a similar proposition if λ1 is a constant in (−1, 1) and λ2 = 1.

Proposition 3.3.8. Let M2 be a constant angle surface immersed inM2(c1) ×M2(c2). Suppose that λ1 is a constant in (−1, 1) and λ2 =1. Then we can find an adapted frame {e1, e2, ξ1, ξ2} such that fe1 =cos(2θ)e1, fe2 = e2, tξ1 = − cos(2θ)ξ1 and tξ2 = −ξ2, where cos(2θ) =λ1and such that the shape operators Sξ1 and Sξ2 take the following formwith respect to the orthonormal frame {e1, e2} :

Sξ1 =(

0 00 µ1

), Sξ2 =

(µ2 00 0

), (3.16)

for some functions µ1 and µ2 on M2. Moreover the Levi-Civitaconnection ∇ of M2 and the normal connection ∇⊥ of M2 in M2(c1)×M2(c2) are given by

g(∇Xe1, e2) = − cot(θ)µ1g(X, e2), (3.17)

g(∇⊥Xξ1, ξ2) = − cot(θ)µ2g(X, e1). (3.18)


K = c1 cos2(θ), (3.19)

and the normal curvature K⊥ is equal to 0.

Finally we consider the case for which λ1 = cos(2θ1) and λ2 = cos(2θ2)are constant, λ2 − λ1 > 0 and λ1, λ2 ∈ (−1, 1). Let {e1, e2, ξ1, ξ2} bean adapted frame such that fei = cos(2θi)ei and tξi = − cos(2θi)ξi fori = 1, 2. Using equations (2.18) and (2.20) and by similar reasoning asbefore, we obtain the next proposition.

Proposition 3.3.9. Let M2 be a surface immersed in M2(c1)×M2(c2).Suppose that λ1 and λ2 are constants in (−1, 1) and λ2 − λ1 > 0. Thenwe can find an adapted orthonormal frame {e1, e2, ξ1, ξ2} such that fei =cos(2θi)ei and tξi = − cos(2θi)ξi, where cos(2θi) = λi for i = 1, 2 and


such that the shape operators Sξ1 and Sξ2 take the following form withrespect to the orthonormal frame {e1, e2}:

Sξ1 =(

0 00 µ1

), Sξ2 =

(µ2 00 0

),

for some functions µ1 and µ2 on M2. Moreover the Levi-Civitaconnection ∇ of M2 and the normal connection ∇⊥ of M2 in M2(c1)×M2(c2) are given by:

g(∇Xe1, e2) = cos(θ1) sin(θ1)µ1g(X, e2) + cos(θ2) sin(θ2)µ2g(X, e1)cos2(θ1)− cos2(θ2) ,

(3.20)

g(∇⊥Xξ1, ξ2) = cos(θ1) sin(θ1)µ2g(X, e1) + cos(θ2) sin(θ2)µ1g(X, e2)cos2(θ1)− cos2(θ2) .

(3.21)


K = c1 cos2(θ1) cos2(θ2) + c2 sin2(θ1) sin2(θ2), (3.22)

and the normal curvature K⊥ equals | c1+c24 | sin(2θ1) sin(2θ2).

3.4 Existence results

We will need the following existence results in the next section.

Proposition 3.4.1. Let c1, c2 ∈ R, not both 0, θ1, θ2 ∈ (0, π2 ) withθ1 > θ2. Define constants a1, a2, A1 and A2 by

a1 = sin(2θ1)cos(2θ1)− cos(2θ2) , a2 = sin(2θ2)

cos(2θ2)− cos(2θ1) ,

A1 = (cos2(θ2)− cos2(θ1))(c1 cos2(θ2)− c2 sin2(θ2)),and

A2 = (cos2(θ1)− cos2(θ2))(c1 cos2(θ1)− c2 sin2(θ1)).Let µ1 = µ1(u, v) and µ2 = µ2(u, v) be real-valued functions defined on asimply connected open subset of R2 which satisfy

− a1√µ2

2 +A2= (µ1)uµ2

1 +A1, − a2√

µ21 +A1

= (µ2)vµ2

2 +A2. (3.23)

EXISTENCE RESULTS 45

Then the Riemannian manifold M2 = (U, g) with the Riemannian metricg = du2

µ22+A2

+ dv2

µ21+A1

is a surface of constant curvature c1 cos2(θ1) cos2(θ2)+c2 sin2(θ1) sin2(θ2). Define now on the vector bundle TU a second metricg by sin2(2θ1) du2

µ22+A2

+sin2(2θ2) dv2

µ21+A1

and denote this Riemannian vectorbundle by T⊥M2. Let f : TM2 → TM2, t : T⊥M2 → T⊥M2, andh : TM2 → T⊥M2 be, respectively, (1, 1) tensors over M2 defined by(

cos(2θ1) 00 cos(2θ2)

),

(− cos(2θ1) 0

0 − cos(2θ2)

),

(1 00 1

),

(3.24)with respect to {∂u, ∂v} and {∂u, ∂v}, where ∂u, ∂v ∈ T⊥M2 and dual tothe forms du and dv. Finally define a symmetric (1, 2) tensor σ withvalues in T⊥M2 and a connection ∇⊥ on T⊥M2 compatible with themetric by

σ(∂u, ∂u) =µ2√µ2

1 +A1

sin(2θ2)(µ22 +A2)

h∂v, σ(∂u, ∂v) = 0,

σ(∂v, ∂v) =µ1√µ2

2 +A2

sin(2θ1)(µ21) +A1

h∂u,

(3.25)

∇⊥∂uh∂u = −µ2(µ2)uµ2

2 +A2h∂u +

a1 sin(2θ1)µ2√µ2

1 +A1

sin(2θ2)(µ22 +A2)

h∂v,

∇⊥∂uh∂v = ∇⊥∂vh∂u = h(∇∂u∂v) = h(∇∂v∂u),

∇⊥∂vh∂v =a2 sin(2θ2)µ1

√µ2

2 +A2

sin(2θ1)(µ21 +A1)

h∂u −µ1(µ1)vµ2

1 +A1h∂v,

where ∇ is the Levi-Civita connection of M2. Then (M2, g, T⊥M2, g, σ,∇⊥, f, h, t) satisfies the compatibility equations of M2(c1)×M2(c2) andhence there exists an isometric immersion of M2 in M2(c1)×M2(c2).Moreover this surface is a constant angle surface and is unique up toisometries of M2(c1)×M2(c2).

Proof. From (3.23) and a direct computation we know that theRiemannian metric g has constant curvature c1 cos2(θ1) cos2(θ2) +


c2 sin2(θ1) sin2(θ2) and the Levi-Civita connection satisfies

∇∂u∂u = −µ2(µ2)uµ2

2 +A2∂u −

a2µ2√µ2

1 +A1

µ22 +A2

∂v,

∇∂u∂v = ∇∂v∂u = a2µ2√µ2

1 +A1∂u + a1µ1√

µ22 +A2

∂v,

∇∂v∂v = −a1µ1

√µ2

2 +A2

µ21 +A1

− µ1(µ1)vµ2

1 +A1∂v.

We have already defined a second metric g on the vector bundle TU ,and denoted this Riemannian vector bundle by T⊥M2, together witha connection ∇⊥ that is compatible with this metric. Let f, t and hbe (1, 1) tensors as defined above and σ the symmetric (1, 2) tensordefined by (3.25). By direct straightforward computations we can seethat (M2, g, T⊥M2, g,∇⊥, σ, f, h, t) satisfies the compatibility equationsfor M2(c1)×M2(c2). Hence there exists an isometric immersion of M2

into M2(c1) ×M2(c2). Moreover, we can deduce from equation (2.46)that M2 is a constant angle surface in M2(c1) ×M2(c2). We can alsoconclude from Theorem (2.3.4) that this immersion with the given secondfundamental form and normal connection is unique up to rigid motionsof M2(c1)×M2(c2).

The next two propositions can be proven analogously as the previousone.

Proposition 3.4.2. Let c1, c2 ∈ R, not both 0, θ1, θ2 ∈ (0, π2 ) withθ1 > θ2. Define constants a1, a2, A1 and A2 as in Proposition 3.4.1.Moreover we suppose that A1 < 0. Let µ = µ(u, v) and G = G(u, v) bereal-valued functions which satisfy

−a2√G = (µ)v

µ2 +A2, a1

√−A1

µ2 +A2= Gu

2G. (3.26)

Then the Riemannian manifold M2 = (U, g) with the Riemannian metricg = du2

µ2+A2+Gdv2 is a surface of constant curvature c1 cos2(θ1) cos2(θ2)+

c2 sin2(θ1) sin2(θ2). Define now on the vector bundle TU a second metricg by sin2(2θ1) du2

µ22+A2

+ sin2(2θ2)Gdv2 and denote this Riemannian vector


bundle by T⊥M2. Let f : TM2 → TM2, t : T⊥M2 → T⊥M2, and h :TM2 → T⊥M2 be (1, 1) tensors over M2 as defined above in Proposition3.4.1. Finally define a symmetric (1, 2) tensor σ with values in T⊥M2

and a connection ∇⊥ on T⊥M2 compatible with the metric by

σ(∂u, ∂u) = µ2

sin(2θ2)(µ22 +A2)

√Gh∂v, σ(∂u, ∂v) = 0,

σ(∂v, ∂v) =G√−A1(µ2

2 +A2)sin(2θ1) h∂u,

∇⊥∂uh∂u = −µ2(µ2)uµ2

2 +A2h∂u + a1 sin(2θ1)µ2

sin(2θ2)(µ22 +A2)

√Gh∂v,

∇⊥∂uh∂v = ∇⊥∂vh∂u = h(∇∂u∂v) = h(∇∂v∂u),

∇⊥∂vh∂v =a2 sin(2θ2)µ1

√−A1(µ2

2 +A2)Gsin(2θ1) h∂u + Gv

2Gh∂v,


Proposition 3.4.3. Let c1, c2 ∈ R, not both 0, θ1, θ2 ∈ (0, π2 ) withθ1 > θ2. Define constants a1, a2, A1 and A2 as in Proposition 3.4.1.Moreover we suppose that A1, A2 < 0. Let E = E(u, v) and G = G(u, v)be positive real-valued functions which satisfy

a2√−A2G = Ev

2E , a1√−A1E = Gu

2G. (3.27)

Then the Riemannian manifold M2 = (U, g) with the Riemannian metricg = Edu2 +Gdv2 is a surface of constant curvature c1 cos2(θ1) cos2(θ2) +c2 sin2(θ1) sin2(θ2). Define now on the vector bundle TU a second metricg by sin2(2θ1)Edu2 + sin2(2θ2)Gdv2 and denote this Riemannian vectorbundle by T⊥M2. Let f : TM2 → TM2, t : T⊥M2 → T⊥M2, andh : TM2 → T⊥M2 be (1, 1) tensors over M2 as defined in Proposition3.4.1. Finally define a symmetric (1, 2) tensor σ with values in T⊥M2


and a connection ∇⊥ on T⊥M2 compatible with the metric by

σ(∂u, ∂u) =√−A2E

sin(2θ2)√Gh∂v, σ(∂u, ∂v) = 0,

σ(∂v, ∂v) =√−A1G

sin(2θ1)√Eh∂u,

∇⊥∂uh∂u = Eu2Eh∂u + a1 sin(2θ1)

√−A2E√

Gh∂v,

∇⊥∂uh∂v = ∇⊥∂vh∂u = h(∇∂u∂v) = h(∇∂v∂u),

∇⊥∂vh∂v = a2 sin(2θ2)√−A1G√

Eh∂u + Gv

2Gh∂v,


Remark 3.4.4. We would like to make some remarks on the equations(3.23). Suppose that A1, A2 > 0. If we define functions θi such thatµi =

√Ai cot(θi), then equations (3.23) are equivalent to

−a1

√A1A2

sin(θ2) = (θ1)u,

−a2

√A2A1

sin(θ1) = (θ2)v.

(3.28)

Differentiating the first equation with respect to v and second equationwith respect to u gives us

(θ1)uv = −a1

√A1A2

cos(θ2)(θ2)v = a1a2 cos(θ2) sin(θ1), (3.29)

(θ2)vu = −a2

√A2A1

cos(θ1)(θ1)u = a1a2 cos(θ1) sin(θ2). (3.30)


The operations (3.29) + (3.30) and (3.29)− (3.30) yield

(θ1 + θ2)uv = a1a2 sin(θ1 + θ2),

(θ1 − θ2)uv = a1a2 sin(θ1 − θ2).

Hence we have found a correspondence between some constant anglesurfaces in M2(c1)×M2(c2) and the Sine-Gordon equation. We wouldlike to remark that the equations (3.28) are the Bäcklund transformationsfor this Sine-Gordon equation. So we obtain a big range of surfaces withconstant angle in M2(c1) × M2(c2). With similar reasoning we finda correspondence with the Sinh-Gordon equation and some constantangle surfaces in M2(c1) ×M2(c2) if A1, A2 < 0. We define functionsθi : U → (0,∞), such that µi =

√−Ai coth(θi), then equations (3.23)

are equivalent to

−a1

√A1A2

sinh(θ2) = (θ1)u,

−a2

√A2A1

sinh(θ1) = (θ2)v.

Differentiating the first equation with respect to v and second equationwith respect to u gives us

(θ1)uv = −a1

√A1A2

cosh(θ2)(θ2)v = a1a2 cosh(θ2) sinh(θ1), (3.31)

(θ2)vu = −a2

√A2A1

cosh(θ1)(θ1)u = a1a2 cosh(θ1) sinh(θ2). (3.32)

The operations (3.31) + (3.32) and (3.31)− (3.32) yield

(θ1 + θ2)uv = a1a2 sinh(θ1 + θ2),

(θ1 − θ2)uv = a1a2 sinh(θ1 − θ2).

Finally we suppose that A1 < 0 and A2 > 0. We define functionsθ1 : U → (0,∞), such that µ1 =

√−A1 coth(θi), and θ2 : U → (0, π),


such that µ2 =√A2 cot(θ2), then equations (3.23) are equivalent to

−a1

√−A1A2

sin(θ2) = (θ1)u,

−a2

√−A2A1

sinh(θ1) = (θ2)v.

Differentiating the first equation with respect to v and the second equationwith respect to u gives us

(θ1)uv = −a1

√−A1A2

cos(θ2)(θ2)v = a1a2 cos(θ2) sinh(θ1), (3.33)

(θ2)vu = −a2

√−A2A1

cosh(θ1)(θ1)u = a1a2 cosh(θ1) sin(θ2). (3.34)

The operations (3.33) + i(3.34) and (3.33)− i(3.34) yield

(θ1 + iθ2)uv = a1a2 sinh(θ1 + iθ2),

(θ1 − iθ2)uv = a1a2 sinh(θ1 − iθ2).

3.5 Main Theorems

In this final section we will classify all the constant angle surfaces inM2(c1)×M2(c2). We split the classification in several subcases. Supposefirst that λ1 = −1 and λ2 = cos(2θ) is a constant in (−1, 1). We willprove the following theorem.

Theorem 3.5.1. A surface M2 isometrically immersed in M2(c1) ×M2(c2) is a constant angle surface with angles θ and π

2 if and only if theimmersion ψ : R2 : (u, v)→ ψ(u, v) is locally given by

(f(v), cos(√c2 sin(θ)v)f(u) + sin(

√c2 sin(θ)v)f(u)× f ′(u)),

if c2 > 0 and where f is a curve in M2(c1) of constant speed cos(θ) andf is a unit speed curve in M2(c2),

(f(v), cosh(√|c2| sin(θ)v)f(u) + sinh(

√|c2| sin(θ)v)f(u) � f

′(u)),

MAIN THEOREMS 51

if c2 < 0 and where f is a curve in M2(c1) of constant speed cos(θ) andf is a unit speed curve in M2(c2),

(f(v), u, sin(θ)v) or (f(v), v sin(θ)f(u) + g(u)), (3.35)

if c2 = 0 and where f is a curve in M2(c1) of constant speed cos(θ),f(u) = (cos(u), sin(u)) and g′(u) = cos(θ)C(u)(− sin(u), cos(u)), whereC is a function on an interval I.

Proof. After a straight-forward computation, one can verify that thesurfaces listed in the theorem are constant angle surfaces in M2(c1)×M2(c2) with angles θ and π

2 .

Conversely, let ψ : M2 →M2(c1)×M2(c2) be a constant angle surface,with λ1 = −1 and λ2 = cos(2θ). Then Proposition 3.3.7 tells us thatwe can find an adapted orthonormal frame {e1, e2, ξ1, ξ2} such thatfe1 = −e1, fe2 = λ2e2, tξ1 = ξ1 and tξ2 = −λ2ξ2 and such that theshape operators associated to ξ1 and ξ2 with respect to e1 and e2 aregiven by

Sξ1 =(

0 00 µ1

), Sξ2 =

(µ2 00 0

),

for some functions µ1 and µ2 on M2. Using (3.13) we obtain that theLevi-Civita connection satisfies

∇e1e1 = tan(θ)µ2e2,

∇e1e2 = − tan(θ)µ2e1, (3.36)

∇e2e1 = 0, (3.37)

∇e2e2 = 0.

From equations (3.36) and (3.37) and the fact that [∂u, ∂v] = 0, wecan deduce that there exist locally coordinates (u, v) on M2 such that∂u = αe1 and ∂v = e2 with

αv = αµ2 tan(θ). (3.38)

Hence the metric takes the form

ds2 = α2du2 + dv2,


and the Levi-Civita connection is given by

∇∂u∂u = αuα∂u − ααv∂v,

∇∂u∂v = ∇∂v∂u = tan(θ)µ2∂u,

∇∂v∂v = 0.

We can also calculate the normal connection ∇⊥ of M2 using (3.14):

∇⊥∂uξ1 = ∇⊥∂uξ2 = 0,

∇⊥∂vξ1 = − tan(θ)µ1ξ2,

∇⊥∂vξ2 = tan(θ)µ1ξ1.

The Codazzi equation gives us now that

(µ1)u = 0, (3.39)

(µ2)v = −µ22 tan(θ)− cos(θ) sin(θ)c2. (3.40)

We immediately see that µ1 is a function that depends only on v. Wesolve now equations (3.38) and (3.40). From equation (3.40) we see thatµ2 must satisfy the following PDE:

(µ2)v = − tan(θ)(c2 cos2(θ) + µ22).

By integration we obtain that µ2 must be equal to−√c2 cos(θ) tan(√c2 sin(θ)v + C(u)) if c2 > 0,0 or 1

tan(θ)v+C(u) if c2 = 0,±√|c2| cos(θ) or

√|c2| cos(θ) tanh(

√|c2| sin(θ)v + C(u)) if c2 < 0,

where C is some function depending on u. Now, solving (3.38) we seethat α equals

D(u) cos(√c2 sin(θ)v + C(u)) if c2 > 0,D(u) or D(u)(tan(θ)v + C(u)) if c2 = 0,D(u) exp (±

√|c2| sin(θ)v) or

D(u) cosh(√|c2| sin(θ)v + C(u)) if c2 < 0,

MAIN THEOREMS 53

where D is some strict positive function depending on u.

We will only consider the case for which c2 > 0. The other cases canbe treated analogously and the results of the other cases are stated inTheorem 3.5.1. So we can consider M2(c1)×M2(c2) as a submanifoldof E5,E6

1 or E6 of codimension 1 or 2 and denote by D the connectionof E5,E6

1 or E6. Hence M2 is an immersed surface in E5,E61 or E6.

Remark now that ξ1, ξ2, which are tangent to M2(c1) ×M2(c2), andξ = (0, 0, ψ4, ψ5, ψ6) are normals ofM2 in E5 if c1 = 0 and that ξ1, ξ2, ξ =(ψ1, ψ2, ψ3, 0, 0, 0) and ξ are normals ofM2 in E6

1 or E6 if c1 6= 0. Moreoverwe have that Fξ1 = ξ1 and hence ξ1 is parallel to the first component ofM2(c1)×M2(c2). One can verify that we have for every X ∈ TpM2,

DX ξ =(I + F

2

)X =

(I + f

2

)X + hX

2 (3.41)

andDXξ =

(I − F

2

)X =

(I − f

2

)X − hX

2 , (3.42)

where F is the product structure of M2(c1) ×M2(c2). Moreover theformulas of Gauss and Weingarten give that:

DXY = ∇XY + σ(X,Y )

−c12 g(

(I + f

2

)X,Y )ξ − c2

2 g((I − f

2

)X,Y )ξ, (3.43)

DXξ1 = −Sξ1X +∇⊥Xξ1, (3.44)

DXξ2 = −Sξ2X +∇⊥Xξ2 −c12 g(hX, ξ2) + c2

2 g(hX, ξ2). (3.45)

In the following we will consider the case for which c1 > 0. The case forwhich c1 ≤ 0 can be treated analogously. Since ∂u = αe1 we find usingequation (3.41) that

(ψ1, ψ2, ψ3, 0, 0, 0)u = D∂u ξ = 0,

and hence we obtain that ψi(u, v) = fi(v) for i = 1, . . . , 3. Analogouslywe find that

(ξ2)i = tan(θ)(ψi)v for i = 1, . . . , 3;

(ξ2)j = − cot(θ)(ψj)v for j = 4, . . . , 6.


We use now the formula of Gauss and the previous equations to find that

(ψj)uu = αuα

(ψj)u − ααv(ψj)v − cot(θ)µ2α2(ψj)v − c2α

2ψj , (3.46)

(ψj)uv = αvα

(ψj)u = tan(θ)µ2(ψj)u, (3.47)

(ψj)vv = −c2 sin2(θ)ψj (3.48)

for j = 4, 5, 6. Integrating equation (3.47), we find that

(ψj)u = cos(√c2v + C(u))Hj(u)

and hence we obtain that

ψj =∫ u

u0cos(√c2v + C(τ))Hj(τ)dτ + Ij(v)

for j = 4, 5, 6 and with Hj and Ij arbitrary functions. Moreover, usingequation (3.48) we find that the functions Ij must satisfy

Ij(v) = Kj cos(√c2 sin θv) + Lj sin(

√c2 sin(θ)v),

where Kj and Lj are constant. We summarize the previous and see thatour immersion ψ is given by

ψ = (f1(v), f2(v), f3(v),(K4 +

∫ u

u0H4(τ) cos(C(τ))dτ

)cos(√c2 sin(θ)v)

+(L4 −

∫ u

u0H4(τ) sin(C(τ))dτ

)sin(√c2 sin(θ)v), . . . ).

We define now the functions

f j(u) = Kj +∫ u

u0Hj(τ) cos(C(τ))dτ,

gj(u) = Lj −∫ u

u0Hj(τ) sin(C(τ))dτ.

MAIN THEOREMS 55

We use now some conditions to find a relation between f(u) =(f4(u), f5(u), f6(u)) and g(u) = (g4(u), g5(u), g6(u)):

g(ψu, ψu) = α2, g(ψv, ψv) = 1, g(ψu, ψv) = 0,

g(ξ1, ψu) = 0, g(ξ1, ψv) = 0, g(ξ1, ξ1) = 1,

g(ξ2, ψu) = 0, g(ξ2, ψv) = 0, g(ξ2, ξ2) = 1,

g(ξ, ψu) = 0, g(ξ, ψv) = 0, g(ξ, ξ) = 1c1,

g(ξ, ψu) = 0, g(ξ, ψv) = 0, g(ξ, ξ) = 1c2,

g(ξ1, ξ2) = 0, g(ξ1, ξ) = 0, g(ξ2, ξ) = 0, g(ξ1, ξ) = 0, g(ξ2, ξ) = 0,

which are equivalent to

3∑i=1

f2i = 1

c1,

3∑j=1

f2j = 1

c2,

3∑j=1

g2j = 1

c2,

3∑j=1

f jgj = 0,3∑j=1

f′jgj = 0,

3∑i=1

(f ′i)2 = cos2(θ),

D2(u) cos2(√c2 sin(θ)v + C(u)) =

3∑j=1

(f ′j)2 cos2(√c2 sin(θ)v)

+(g′j)2 sin2(√c2 sin(θ)v) + 2f ′jg′j cos(

√c2 sin(θ)v) sin(

√c2 sin(θ)v).

(3.49)

From the above equations we see that f(u) = (f1(u), f2(u), f3(u)) andg(u) = (g1(u), g2(u), g3(u)) are curves in M2(c2). Moreover if we changethe u-coordinate such that f is a unit speed curve, which corresponds


to the fact that D2(u) = sec2(C(u)), we see then from the previousequations that g is a curve in M2(c2) that is perpendicular to the vectorsf and f ′. Hence we obtain that g = ±f × f ′ and we can choose thatg = f × f ′. The immersion ψ is then given by

ψ(u, v) = (f(v), cos(√c2 sin(θ)v)f(u) + sin(

√c2 sin(θ)v)f(u)× f ′(u)).

Let us remark that since g = f × f ′, we obtain that g′ = 1√c2

(Jf ′)′ =− κ√

c2f′ and hence we have f ′ · g′ = − κ√

c2. Using equation (3.49), we

obtain that κ√c2

= tan(C(u)).

The case for which λ1 = cos(2θ) and λ2 = 1 can be treated analogouslyas the previous case. We summarize this case in the next theorem:

Theorem 3.5.2. A surface M2 isometrically immersed in M2(c1) ×M2(c2) is a constant angle surface with angles 0 and θ if and only if theimmersion ψ : (u, v) 7→ ψ(u, v) is locally given by

(cos(√c1 cos(θ)u)f(v) + sin(

√c1 cos θ)u)f(v)× f ′(v), f(u)),

if c1 > 0 and where f is a curve in M2(c2) of constant speed sin(θ) andf is a unit speed curve in M2(c1),

(cosh(√|c1| cos(θ)u)f(v)

+ sinh(√|c1| cos(θ)u)f(v) � f ′(v), f(u)),

if c1 < 0 and where f is a curve in M2(c2) of constant speed sin(θ) andf is a unit speed curve in M2(c1),

(cos(θ)u, v, f(u)) or u cos(θ)f(v) + g(v), f(u)), (3.50)

if c1 = 0 and where f is a curve in M2(c2) of constant speed sin(θ),f(v) = (cos(v), sin(v)) and g′(v) = − sin(θ)C(v)(− sin(v), cos(v)), whereC is a function on an interval I.

We consider now the case for which λ1, λ2 ∈ (−1, 1) and λ2− λ1 ≥ 0 andshow the following theorem.

MAIN THEOREMS 57

Theorem 3.5.3. Let M2 be a constant angle surface with θ1, θ2 ∈ (0, π2 ).Then there are two possibilities:

1. M2 is an open part of the surfaces parameterized by

(cos(√

c1c2c1 + c2

u)f(v) + sin(√

c1c2c1 + c2

u) 1cos(θ) f(v)× f ′(v);

cos(√

c1c2c1 + c2

u)f(v) + sin(√

c1c2c1 + c2

u) 1sin(θ)f(v)× f ′(v)),

if c1, c2 > 0,

(cosh(√− c1c2c1 + c2

u)f(v) + sinh(√− c1c2c1 + c2

u) 1cos(θ) f(v) � f ′(v);

cosh(√− c1c2c1 + c2

u)f(v) + sinh(√− c1c2c1 + c2

u) 1sin(θ)f(v) � f

′(v)),

if c1, c2 < 0, where f is a curve in M2(c1) of constant speed cos(θ)and geodesic curvature κ and f is a curve in M2(c2) of constantspeed sin(θ) and geodesic curvature κ, such that κ√

|c1|= κ√

|c2|,

2. a constant angle surface in M2(c1)×M2(c2) given by Proposition3.4.1, 3.4.2 or 3.4.3.

Proof. After a straight-forward computation, one can deduce that thesurfaces listed in the theorem are constant angle surfaces in M2(c1)×M2(c2). Conversely, let us assume that M2 is a constant angle surfacein M2(c1) × M2(c2) with angles θ1, θ2 ∈ (0, π2 ). Suppose first thatθ1 = θ2. Then M2 is a totally geodesic surface in M(c1) ×M(c2) andλ1 = λ2 = − b

a . Hence M2 is locally congruent to (3.7) and (3.8). So weare in the special case of case 1 of the theorem. Let us now suppose thatθ1 6= θ2. Then there is an adapted orthonormal frame {e1, e2, ξ1, ξ2} suchthat fei = cos(2θi)ei and tξi = − cos(2θi)ξi, where cos(2θi) = λi, fori = 1, 2. Moreover we have that the shape operators Sξ1 and Sξ2 have thesame form as (3.12) with respect to {e1, e2}. Moreover the Levi-Civitaconnection is given by

∇e1e1 = sin(θ2) cos(θ2)µ2cos2(θ1)− cos2(θ2)e2,




∇e2e2 = sin(θ1) cos(θ1)µ1cos2(θ2)− cos2(θ1)e1.

We also know the normal connection ∇⊥ of M2 in M2(c1)×M2(c2):

∇⊥e1ξ1 = sin(θ1) cos(θ1)µ2cos2(θ1)− cos2(θ2)ξ2,



∇⊥e2ξ2 = sin(θ2) cos(θ2)µ2cos2(θ2)− cos2(θ1)ξ1.

Using the expressions for the Levi-Civita connection and the normalconnection, we find that the Codazzi equations are given by

e1[µ1] + sin(θ1) cos(θ1)cos2(θ1)− cos2(θ2)µ

21

= (c1 cos2(θ2)− c2 sin2(θ2)) sin(θ1) cos(θ1), (3.51)

e2[µ2] + sin(θ2) cos(θ2)cos2(θ2)− cos2(θ1)µ

22

= (c1 cos2(θ1)− c2 sin2(θ1)) sin(θ2) cos(θ2). (3.52)

Case 1: µ1 = µ2 = 0. From the equations of Codazzi (3.51) and (3.52) weobtain that c1 cos2(θ1)−c2 sin2(θ1) = c1 cos2(θ2)−c2 sin2(θ2) = 0, becauseθ1, θ2 ∈ (0, π2 ) and hence we obtain that cos(2θ1) = cos(2θ2) = c2−c1

c1+c2with c1c2 > 0. Since cos(2θ2) > cos(2θ1) by assumption, we have acontradiction.

Case 2: µ1 = 0, µ2 6= 0. As before, using the equation of Codazzi, weobtain that c1 cos2(θ2) − c2 sin2(θ2) = 0 and hence we obtain that

MAIN THEOREMS 59

cos(2θ2) = c2−c1c1+c2

with c1c2 > 0. Denote in the following θ1 by θ. Wewill work out only the case for which c1 > 0 and c2 > 0. The other casecan be treated analogously. From the expressions for the Levi-Civitaconnection, we find that ∇e2e1 = ∇e2e2 = 0. Let us take now coordinateson M2 with ∂u = αe1 and ∂v = βe2. Using the condition [∂u, ∂v] = 0and the expressions for the Levi-Civita connection we find that

αv =√c1c2

c2 sin2(θ)− c1 cos2(θ)αβµ2, (3.53)

βu = 0. (3.54)

Equation (3.54) implies that, after a change of the u-coordinate, we canassume that β = 1 and hence the metric takes the form

g = α2du2 + dv2,

and so the Levi-Civita connection becomes:

∇∂u∂u = αuα∂u − ααv∂v,

∇∂u∂v = ∇∂v∂u =√c1c2

c2 sin2(θ)− c1 cos2(θ)µ2∂u,

∇∂v∂v = 0.

The equation of Codazzi (3.52) can now be rewritten as

(µ2)v = c1c2c1 cos2(θ)− c2 sin2(θ)

((c1 cos2(θ)− c2 sin2(θ))2

c1 + c2+ µ2

2

).

(3.55)Integrating equations (3.53) and (3.55) we find

µ2 = c1 cos2(θ)− c2 sin2(θ)√c1 + c2

tan(√

c1c2c1 + c2

v + C(u)),

α = D(u) cos(√

c1c2c1 + c2

v + C(u)).

Let us remark that we obtain form previous equation that M2 hasconstant Gaussian curvature c1c2

c1+c2. We denote c1c2

c1+c2by K.


We consider now the surface M2 as a codimension 4 immersed surface inthe Euclidean space E6. By D we will denote the Euclidean connection.We remark that ξ1, ξ2, ξ = (ψ1, ψ2, ψ3, 0, 0, 0) and ξ = (0, 0, 0, ψ4, ψ5, ψ6)are normals of M2 in E6. We still have that the equations (3.41) and(3.42) hold. Moreover the equations of Gauss and Weingarten are givenby

DXY = ∇XY + σ(X,Y )

−c12 g(

(I + f

2

)X,Y )ξ − c2

2 g((I − f

2

)X,Y )ξ, ,

DXξ1 = −Sξ1X +∇⊥Xξ1 −c12 g(hX, ξ1) + c2

2 g(hX, ξ1),

DXξ2 = −Sξ2X +∇⊥Xξ2 −c12 g(hX, ξ2) + c2

2 g(hX, ξ2).

Now applying the formula Gauss and the previous equations we find

D∂u∂u = αuα∂u − ααv∂v + µ2α

2ξ2 − c1 cos2(θ1)α2ξ − c2 sin2(θ1)α2ξ,

(3.56)

D∂u∂v = D∂v∂u = −√K tan(

√Kv + C(u))∂u, (3.57)

D∂v∂v = −K(ξ + ξ), (3.58)where

(ξ2)i =√c1c2

(ψi)v for i = 1, 2, 3,

(ξ2)j = −√c1c2

(ψj)v for j = 4, 5, 6.

Integrating the last two formulas of Gauss, i.e. (3.57) and (3.58), weobtain analogously as before that

ψ(u, v) = (cos(√Ku)f(v) + sin(

√Ku)g(v);

cos(√Ku)f(v) + sin(

√Ku)g(v)),

MAIN THEOREMS 61

where f(v) = (f1(v), f2(v), f3(v)), g(v) = (g1(v), g2(v), g3(v)), f(v) =(f1(v) , f2(v), f3(v)) and g(v) = (g1(v), g2(v), g3(v)) and

fi(v) = Ki +∫ v

v0Hi(τ) cos(C(τ))dτ,

gi(v) = Li −∫ v

v0Hi(τ) sin(C(τ))dτ,

f j(v) = Kj +∫ v

v0Hj(τ) cos(C(τ))dτ,

gj(v) = Lj −∫ v

v0Hj(τ) sin(C(τ))dτ,

for i, j = 1, . . . , 3. Moreover we have the following equations

g(ψu, ψu) = α2, g(ψv, ψv) = 1, g(ψu, ψv) = 0,

g(ξ1, ψu) = 0, g(ξ1, ψv) = 0, g(ξ1, ξ1) = 1,

g(ξ2, ψu) = 0, g(ξ2, ψv) = 0, g(ξ2, ξ2) = 1,

g(ξ, ψu) = 0, g(ξ, ψv) = 0, g(ξ, ξ) = 1c1,

g(ξ, ψu) = 0, g(ξ, ψv) = 0, g(ξ, ξ) = 1c2,

g(ξ1, ξ2) = 0, g(ξ1, ξ) = 0, g(ξ2, ξ) = 0, g(ξ1, ξ) = 0, g(ξ2, ξ) = 0,

which are equivalent to3∑i=1

f2i = 1

c1=

3∑i=1

g2i ,

3∑j=1

f2j = 1

c2=

3∑j=1

g2j ,

3∑i=1

figi = 0 =3∑i=1

f ′i gi,3∑j=1

f jgj = 0 =3∑j=1

f′jgj ,

3∑i=1

((f ′i)2 cos2(

√Kv) + (g′i)2 sin2(

√Kv) + 2f ′i g′i cos(

√Kv) sin(

√Kv)

)= cos2(θ)D2(u) cos2(

√Kv + C(u)),


3∑j=1

((f ′j)2 cos2(

√Kv) + (g′j)2 sin2(

√Kv)

+2f ′jg′j cos(√Kv) sin(

√Kv)

)= sin2(θ)D2(u) cos2(

√K sin(θ)v + C(u)).

We obtain from the above equations that f and g are curves of M2(c1),that f and g are curves of M2(c2). If we change the v-coordinate suchthat f and f have constant speed cos(θ) and sin(θ), which correspondsto the fact that D2(v) = sec2(C(v)), we see then from the previousequations that g = ± 1

cos(θ) f× f′ and g = ± 1

sin(θ)f×f′ and we can choose

g = 1cos(θ) f × f

′ and g = 1sin(θ)f × f

′. From the last two equations we

also deduce that f ′ ·g′cos2(θ) = f

′·g′sin2(θ) and g′ ·g′

cos2(θ) = g′·g′sin2(θ) . This is equivalent

to κ√c1

= κ√c2, where κ and κ are the geodesic curvatures of respectively

f and f . So we obtain the first case of the theorem.

Case 3: µ1 6= 0, µ2 6= 0. Let (u, v) be coordinates on M2 such that ∂u =αe1 and ∂v = βe2. From the expression of the Levi-Civita connectionand the condition [∂u, ∂v] = 0, we obtain

a2µ2 = αvαβ

, (3.59)

a1µ1 = βuαβ

, (3.60)

where a1 and a2 are constants as in Proposition 3.4.1. Using the previousequations, we can rewrite the equations of Codazzi (3.51) and (3.52) asfollows

(α2(µ22 +A2))v = 0,

(β2(µ21 +A1))u = 0,

where A1 and A2 are constants as in Proposition 3.4.1 and hence weobtain that α2(µ2

2 +A2) = C2(u) and β2(µ21 +A1) = C1(v). We have to

consider now several subcases.

Case 3.a.: C1 6= 0, C2 6= 0. After a transformation of the u-coordinateand the v-coordinate we can suppose that α2 = 1

µ22+A2

and β2 = 1µ2

1+A1.

MAIN THEOREMS 63

Substituting this in equations (3.59) and (3.60) we obtain equations (3.23).We can conclude that the isometric immersion ψ is locally congruent tothe surface of Proposition 3.4.1.

Case 3.b.: C1 = 0, C2 6= 0. Since C1 = 0, we have that µ21 +A1 = 0. So

we have that A1 < 0, because µ1 6= 0. We conclude that µ1 = ±√−A1

and without loss of generalization we can suppose that µ1 =√−A1 . After

a transformation of the u-coordinate, we can suppose that α2 = 1µ2

2+A2.

Substituting the last two equations into equations (3.59) and (3.60) weobtain equations (3.26). We can conclude that the isometric immersionψ is locally congruent to the surface of Proposition 3.4.2.

Case3.b.: C1 = C2 = 0. Analogously as before we conclude that ψ islocally congruent to the surface of Proposition 3.4.3.

We end this section with classification of the totally geodesic surfacesof M2(c1) ×M2(c2). We have seen that a totally geodesic surface ofM2(c1)×M2(c2) is a constant angle surfaces of M2(c1)×M2(c2). Wewill use the classification of the constant angle surfaces to give theclassification of the totally geodesic surfaces of M2(c1)×M2(c2).

Theorem 3.5.4. LetM2 be a totally geodesic surface ofM2(c1)×M2(c2).Then there are four possibilities

1. M2 is locally congruent to the immersion given by (3.7) or by (3.8);

2. M2 is a product of two geodesic curves;

3. M2 is an open part of M2(c1)× {p2} or {p1} ×M2(c2);

4. M2 is locally congruent to the first immersion of (3.35), in whichthe curve f is a geodesic curve of M2(c1) if c2 = 0, or to the firstimmersion of (3.50), in which the curve f is a totally geodesiccurve of M2(c2) if c1 = 0.

Proof. Proposition 3.3.4 tells us that a totally geodesic surface ofM2(c1) ×M2(c2) is a constant angle surface. Moreover in the proofof Proposition 3.3.4, we have seen that there are four possible situations.In the first case we have seen that the angle functions λ1 and λ2 areequal and have value − b

a , in which a = c1+c24 , b = c1−c2

4 and c1c2 > 0.We have classified these totally geodesic surfaces in Proposition 3.3.5


and showed that they are locally congruent to (3.7) if c1, c2 > 0 or to(3.8) if c1, c2 < 0. The second case says that the angle functions areequal to ±1. If the angle functions have opposite sign then one can easyshow that the surface is a Riemannian product of curves of M2(c1) andM2(c2) and that this surface is totally geodesic if and only if both curvesare geodesic curves of M2(c1) and M2(c2). If both angle functions havethe same sign then one can easily deduce that the surface is an openpart of M2(c1)× {p2} if the angle functions are equal to 1 or an openpart of {p1} ×M2(c2) if the angle functions are equal to −1. The thirdcase in the proof tells us that one of the angle functions is ±1 and theother angle function is equal to − b

a . We show that in this case thereexist only totally geodesic surfaces in the case that c1 = 0 or c2 = 0.Suppose therefore that c1 6= 0 and c2 6= 0. Remark that − b

a 6= ±1,because c1 6= 0 and c2 6= 0. In Propositions 3.3.7 and 3.3.8 we haveshowed that the curvature of the surface is c2 sin2(θ) if one of the anglefunctions is −1 or c1 cos2(θ) if one of the angle functions is 1. So if theother angle function is equal to − b

a , then we obtain that in both casesthe curvature is equal to c1c2

c1+c2. But the surface is totally geodesic and

hence we obtain form equations (3.13) and (3.17) that the surface is flat.Finally we obtain that c1c2

c1+c2= 0 and hence c1 = 0 or c2 = 0. This is

of course a contradiction, because we have assumed that c1 6= 0 andc2 6= 0. Hence we obtain that in the third case c1 = 0 or c2 = 0 andso the angle functions in this case are ±1. This brings us back to thesecond case and hence we are finished. In the fourth case we have thatone of the angle functions is equal to 1 and the other angle function is aconstant in [−1, 1] if c1 = 0 or that one of the angle functions is equalto −1 and the other angle functions is a constant in [−1, 1] if c2 = 0.We can suppose that the other angle function is a constant in (−1, 1).We can easily deduce from the classification theorems 3.5.1 and 3.5.2,that M2 is indeed locally congruent to the first immersion of (3.35), inwhich the curve f is a geodesic curve of M2(c1) if c2 = 0, or to the firstimmersion of (3.50), in which the curve f is a totally geodesic curve ofM2(c2) if c1 = 0.

Remark. The special case when c1 = c2 = 2 of Theorem 3.5.4 was alsoconsidered in the paper [9] where totally geodesic surfaces in Qn (inparticular, in Q2 = S2(2) × S2(2)) were classified. In particular, theyproved that a totally geodesic surface of Qn is one of the following threekinds:

MAIN THEOREMS 65

1. a totally geodesic totally real surface,

2. a totally geodesic complex surface,

3. a totally geodesic surface of curvature 1/5 in Qn which is neithertotally real nor complex. This case occurs only when n ≥ 3.

Since the third case doesn’t occur for n = 2, this is consistent with ourresults. It is interesting to remark that the third case was missing in [10]and [11]. This was remarked by S. Klein in [36], who didn’t notice thatthe missing case did occur in the earlier paper [9].

Chapter 4

On extrinsic symmetries ofhypersurfaces in productspaces

4.1 Introduction

Given a Riemannian manifold, it is an interesting problem to studyand classify its hypersurfaces endowed of special symmetry properties.Natural candidates for this study are totally geodesic (more in general,totally umbilical), parallel and semi-parallel hypersurfaces. Such a studyenriches our understanding of the geometry of the ambient space andpermits a comparison with analogous results obtained in different ambientspaces.

Real space forms are clearly the first candidates for the study of theirhypersurfaces, since they are the ambient spaces with the simplest formof the curvature tensor. Semi-parallel hypersurfaces in a space form havebeen classified in [17] and [18] .

Besides space forms, the Riemannian manifolds endowed of the simplestcurvature tensor are the remaining conformally flat symmetric spaces,that is, the Riemannian products Sn(c) × R, Hn(c) × R and Sp(c) ×Hn−p(−c). It is then natural to consider these ambient spaces and to tryto classify their hypersurfaces endowed of special extrinsic symmetriesproperties. This study was started by Van der Veken and Vrancken in [58],

66

EXTRINSIC SYMMETRIES 67

where hypersurfaces of Sn(c)× R have been investigated. In the presentchapter we will not only study totally umbilical, parallel and semi-parallelhypersurfaces in product spaces Hn(c)×R and Sp+1(c)×Hn−p(−c), butalso in M(c1)×M(c2) for which c1c2 6= 0.

The chapter is organized in the following way. In the first section we willrecall the definitions of extrinsic symmetries of hypersurfaces and thegeometric interpretations of these symmetries. In section 4.3 we will recallsome definitions and notations about multiple warped product spaces. Inthe next section we will study the extrinsic symmetries inHn(c)×R, wherewe classify totally umbilical, semi-parallel and parallel hypersurfaces andexplain their relationship with the rotation hypersurfaces of Hn(c)× R,which were introduced in [20] together with the rotation hypersurfacesof Sn(c)×R. It is worthwhile to note that with respect to the analogousstudy on the ambient space Sn(c)× R, more examples arise. In the lastsection we will classify the totally umbilical, semi-parallel and parallelhypersurfaces of M(c1)×M(c2) with c1c2 6= 0. We would like to notethat in this case no new examples arise. The results of first part arepublished in [5].

4.2 Extrinsic symmetries

Let us reminisce first the extrinsic symmetries conditions for hy-persurfaces. We begin with totally geodesic hypersurfaces. Thisare hypersurfaces for which the second fundamental form h vanishesidentically. Geometrically these hypersurfaces are characterized by theproperty that geodesic curves on the hypersurface are also geodesic curvesin the ambient space. The geodesic hypersurfaces of the Euclidean spaceEn+1 are open parts of the Euclidean space En.

We call a hypersurface parallel if ∇h = 0, i.e. the covariant derivativeof the second fundamental form vanishes identically. It is easy to seethat totally geodesic hypersurfaces are parallel hypersurfaces. Parallelsurfaces in the Euclidean space E3 were first studied by Kagan in [34].He showed that these surfaces consists of planes E2, spheres S2 and roundcylinders S1 × E1. The classification of parallel hypersurfaces in realspace forms can be found in [39] and [48].Theorem 4.2.1. A parallel hypersurface of a real space form is one ofthe following:

68 ON EXTRINSIC SYMMETRIES OF HYPERSURFACES IN PRODUCT SPACES

1. in En+1: an open part of a hypersphere Sn(c1), of a hyperplane En,or of a Riemannian product Sk(c1)× En−k,

2. in Sn+1(c): an open part of a Sn(c1) or of a Riemannian productSk(c1)× Sn−k(c2), with 1

c1+ 1

c2= 1

c ,

3. in Hn+1(c): an open part of a Hn(c1), of a Riemannian productHk(c1)× Sn−k(c2), with 1

c2− 1

c1= 1

c , or of a horosphere.

The general classification of parallel submanifolds, i.e. submanifolds forwhich the second fundamental form is parallel, in the Eucliddean spaceswas given in [28]. In [29], Ferus characterized parallel submanifolds asthe submanifolds for which at every point there exists a neighbourhoodwhich is invariant under the reflection of the Euclidean space with respectto the normal space at this point. This characterization was generalizedto ambient spaces with constant curvature by Strübing in [53].

Subsequently parallel hypersurfaces were generalized to a more generalclass of semi-parallel hypersurfaces by Deprez in [17]. This happens in thesame sense as the locally symmetric manifolds, which are characterizedby the property ∇R = 0. The locally symmetric manifolds are namelygeneralized to semi-symmetric manifolds. This are Riemannian manifoldswhich are characterized by the integrability condition R ·R = 0. We callnow a hypersurface semi-parallel if R · h = 0, i.e.

R(X,Y ) · h(X1, X2) = −h(R(X,Y )X1, X2)− h(X1, R(X,Y )X2) = 0.

One can show that R · h = 0 is the integrability condition of ∇h = 0.In [17], the classification of semi-parallel hypersurfaces in the Euclideanspace is given. We can find the classification of semi-parallel hypersurfacesin real space forms in [19].

Theorem 4.2.2. Let M be semi-parallel hypersurface of dimension n inM(c). Then M is one of the following types:

1. M is flat if c = 0;

2. M is flat if c 6= 0 and n = 2;

3. M is parallel;

4. M is a round cone or a product of a round cone and a linearsubspace of En+1 if c = 0;

WARPED PRODUCT METRICS 69

5. There exists a totally geodesic surface M(c), and a vector u inthe linear subspace R3 of Rn+2, containing M(c), such that M isa rotation hypersurface whose profile curve is a u−helix lying inM(c), and whose axis is u⊥.

In [16], Deprez gave the definition of a semi-parallel submanifold. Inthe same paper he studied semi-parallel surfaces of the Euclidean. Theclassification of all semi-parallel submanifolds in the Euclidean space isnot known. In [23], Dillen and Nölker gave a complete description of allsemi-parallel submanifolds with flat normal bundle in real space forms.They showed that a semi-parallel submanifold of a space form withflat normal connection is a multi-rotation submanifold with appropriate"axes" whose "profile" is a submanifold with flat normal bundle satisfyingthe helix property. For more details we would like to refer to a surveyarticle [41] of Lumiste.

Finally we define the totally umbilical hypersurfaces. A hypersurface istotally umbilical if the second fundamental form h is a scalar multiple ofthe metric g at every point, i.e. h(X,Y ) = λg(X,Y ) for some functionλ on the hypersurface.

4.3 Warped product metrics

In this short section we will recall the definition and the properties ofthe Riemannian warped product. Let us take Riemannian manifolds(F0, g0), . . . , (Fm, gm) and consider the product manifold M = F0 ×F1 ×· · · × Fm. Denote by πi : M → Fi the canonical projections, Li theproduct foliation of M induced by Fi and TLi the set of lifts of vectorfields of Fi for i ∈ {0, . . . ,m}. Let bi : F0 → (0,∞) be smooth functionsfor i ∈ {1, . . . ,m}. The Riemannian warped product is the productmanifold furnished with the metric g = g0⊕ b2

1g1⊕ · · ·⊕ b2mgm defined by

g = π∗0(g0) + (b1 ◦ π0)2π∗1(g1) + · · ·+ (bm ◦ π0)2π∗1(gm).

We call the functions bi : F0 → (0,∞) the warping functions. EachRiemannian manifold (Fi, gi) is called a fiber manifold for i ∈ {1, . . . ,m}.The Riemannian manifold (F0, g0) is the base manifold of the Riemannianwarped product manifold M . Let us remark that M is a Riemannianproduct manifold if all the functions bi ≡ 1.


In the next sections we will need the expression of the Riemanniancurvature tensor of a warped product in terms of the base manifold andthe fiber manifold. This is given in the next proposition.

Proposition 4.3.1. Let M = F0 ×b1 F1 × · · · ×bm Fm be a Riemannianwarped product manifold with metric g = g0 ⊕ b2

1g1 ⊕ · · · ⊕ b2mgm. Also

let X,Y, Z ∈ TL0 and V ∈ TLi,W ∈ TLj and U ∈ TLk. Then we havethat the curvature tensor R of M is given by

1. R(X,Y )Z = RF0(X,Y )Z,

2. R(V,X)Y = −HbiF0

(X,Y )bi

V,

3. R(X,V )W = R(V,W )X = R(V,X)W = 0 if i 6= j,

4. R(X,Y )V = 0,

5. R(V,W )X = 0 if i = j,

6. R(V,W )U = 0 if i = j and i, j 6= k,

7. R(U, V )W = −g(V,W )gF0 (gradF0bi,gradF0bk)bibk

U if i = j and i, j 6= k,

8. R(X,V )W = −g(V,W )bi∇F0X (gradF0bi) if i = j,

9. R(V,W )U = RFi(V,W )U −‖gradF0bi‖

2F0

b2i

(V ∧g W )U if i, j = k,

where gradF0 is the gradient of the Riemannian manifold (F0, g0) andHF0 is the Hessian of the Riemannian manifold (F0, g0).

4.4 Extrinsic symmetries in Hn × R

4.4.1 Hypersurfaces of Hn × R

We recall first how the Riemannian product Hn×R is immersed in Ln+2

as a hypersurface and we recall the equations that are necessary to findthe totally uimbilical, semi-parallel and parallel hypersurfaces. In thissection we will use the formalism of Daniel, introduced in [14], in order

EXTRINSIC SYMMETRIES IN HN × R 71

to study the hypersurfaces in Hn × R. Let Ln+2 be the flat Lorentzianspace of dimension n+ 2, i.e.

Ln+2 = (Rn+2,−dx21 + dx2

2 + . . .+ dx2n+2)

and define Hn × R as the following subset of Ln+2, equipped with theinduced metric:

Hn×R = {(x1, . . . , xn+2) | −x21 +x2

2 + · · ·+x2n+1 = −1, x1 > 0}. (4.1)

We know that Hn × R is the Riemannian product of the hyperbolicspace Hn of constant sectional curvature −1 and the real line and thatξ = (x1, . . . , xn+1, 0) is a normal vector field on Hn×R in Ln+2, satisfying〈ξ, ξ〉 = −1. We denote by XHn the projection of a vector field X tangentto Hn × R onto the tangent space of Hn, i.e.

XHn = X − 〈X, ∂n+2〉∂n+2.

Using the formula of Gauss for isometric immersions in semi-Riemannianmanifolds, we find that the Levi-Civita connection ∇ of Hn × R is givenby

∇XY = DXY − 〈XHn , YHn〉ξ, (4.2)

where X,Y are vector fields tangent to Hn ×R and D is the Levi-Civitaconnection of Ln+2. Consequently we obtain that the Riemann curvaturetensor R of Hn × R is determined by

〈R(X,Y )Z,W 〉 = 〈XHn , ZHn〉〈YHn ,WHn〉 − 〈YHn , ZHn〉〈XHn ,WHn〉.(4.3)

Consider a hypersurface x : M → Hn × R with unit normal N . Denoteby T the projection of the vector field ∂n+2 onto the tangent space to Mand denote by θ a function M such that cos θ = 〈N, ∂n+2〉. So we obtainthat ∂n+2 = T + cos θN along M . Using these notations, the equationsof Gauss and Codazzi are given in terms of T and cos(θ) as follows:

R(X,Y )Z = (SX ∧ SY )Z − ((X ∧ Y )Z + 〈Y, T 〉(T ∧X)Z

− 〈X,T 〉(T ∧ Y )Z), (4.4)

(∇Y S)X − (∇XS)Y = − cos(θ)(X ∧ Y )T, (4.5)


where X,Y, Z are vector fields tangent to M and S is the shape operatorassociated to N . Moreover, by using the fact that ∂t is a parallel vectorfield in Hn × R, i.e. ∇X∂t = 0 for every X ∈ T (Hn × R), we obtain

∇XT = νSX, (4.6)

X(cos(θ)) = −〈SX, T 〉. (4.7)

These equations appear in the existence and uniqueness theorem forisometric immersions of hypersurfaces of Hn × R, see Theorem 2.1.2.

4.4.2 Rotation hypersurfaces in Hn × R

We will now recall the definition of a rotation hypersurface of Hn ×R, as proposed in [20]. This definition of rotation hypersurfaces isgeneralization of the definition of rotation hypersurfaces introduced in[26]. Consider a three-dimensional subspace P 3 of Ln+2 containing thexn+2-axis. Then (Hn ×R) ∩ P 3 = H1 ×R. Let P 2 be a two-dimensionalsubspace of P 3, also containing the xn+2-axis. Let I denote the groupof isometries of Ln+2, which leave Hn × R invariant and which leavethe subspace P 2 pointwise fixed. Let α be a curve in H1 × R whichdoes not intersect P 2. Then the rotation hypersurface M of Hn × Rwith profile curve α and axis P 2 is defined as the I-orbit of α. Fromthe definition it follows that the velocity vector of α is proportional toT , unless α lies in a plane orthogonal to ∂n+2, in which case T = 0.We will consider three cases, depending on whether P 2 is Lorentzian,Riemannian or degenerate. Without loss of generality, in the sequelwe will assume that P 3 is spanned by {∂1, ∂n+1, ∂n+2} and that P 2 isspanned by {∂1, ∂n+2} (Lorentzian), by {∂n+1, ∂n+2} (Riemannian) or by{en+1, en+2} (degenerate) where {e1, . . . , en+2} is a pseudo-orthonormalbasis of Ln+2 with

e1 = 1√2

(∂1 + ∂n+1) , en+1 = 1√2

(∂1 − ∂n+1) , ek = ∂k,

for k ∈ {2, . . . , n, n+ 2}. It was proven in [20] that there exists a localorthonormal frame {f1, . . . , fn} on M , with T = ‖T‖f1, such that theshape operator S takes the following diagonal form with respect to


{f1, . . . , fn}:

S =

λ

µ. . .

µ

.The principal curvatures λ and µ are constant on orbits and can becomputed explicitly as follows. Assume first that P 2 is either Riemannianor Lorentzian. If the profile curve α is not a vertical line in H×R, it canbe locally parameterized as α(s) = (cosh(s), 0, . . . , 0, sinh(s), a(s)) andwe have that

λ = − a′′(s)(1 + a′(s)2)

32, µ = −a

′(s) coth(s)(1 + a′(s))

12

if P 2 = span{∂1, ∂n+2}

(4.8)and that

λ = − a′′(s)(1 + a′(s)2)

32, µ = −a

′(s) tanh(s)(1 + a′(s))

12

if P 2 = span{∂n+1, ∂n+2}.

(4.9)If α is a vertical line α(s) = (cosh(c), 0, . . . , 0, sinh(c), s), with c ∈ R, wehave that

λ = 0, µ = − coth(c) if P 2 = span{∂1, ∂n+2} (4.10)

and that

λ = 0, µ = − tanh(c) if P 2 = span{∂n+1, ∂n+2}. (4.11)

Next, assume that the axis of the rotation hypersurface is degenerate. Ifthe profile curve is not a vertical line in Hn × R, it can be locallyparameterized as α(s) = (s, 0, . . . , 0,− 1

2s , a(s)) with respect to thepseudo-orthonormal basis {e1, . . . , en+2}, and we obtain that

λ = −sa′(s) + s2a′′(s)

(1 + s2a′(s))32, µ = − sa′(s)

(1 + s2a′(s))12. (4.12)

If α is a vertical line α(s) = (c, 0, . . . , 0,− 12c , s), with c ∈ R, we obtain

λ = 0, µ = −1. (4.13)

Finally, we also mention the following characterization of rotationhypersurfaces in Hn × R.


Theorem 4.4.1 ([20]). Let n ≥ 3 and let f : M → Hn × R be ahypersurface with shape operator

S =

λ

µ. . .

µ

,with λ 6= µ and suppose ST = λT . Assume moreover that there isa functional relation λ(µ). Then M is an open part of a rotationhypersurface.

4.4.3 Totally Umbilical hypersurfaces in Hn × R

In this section, we give a complete classification of totally umbilicalhypersurfaces in Hn × R. We remark that Van der Veken has classifiedtotally umbilical surfaces in S2 × R and H2 × R, by means of an explicitparametrization, in [57]. Independently, another description of the samefamily of surfaces was obtained in [51]. Moreover Van der Veken andVrancken classified totally umbilical hypersurfaces in Sn × R in [58]. Wewill see that in the case of Hn × R, more families of totally umbilicalhypersurfaces arise.

First, we classify totally geodesic hypersurfaces. The proof of thefollowing theorem is based on the equation of Codazzi and is completelyanalogous to the proof of Theorem 3 in [58].

Theorem 4.4.2. Let M be a totally geodesic hypersurface of Hn × R.Then M is an open part of a hypersurface Hn × {t0} for t0 ∈ R, or ofa hypersurface Mn−1 × R, with Mn−1 a totally geodesic hypersurface ofHn.

Proof. Let M be a totally geodesic hypersurface of Hn × R. Usingequation (4.5) of Codazzi we obtain that there are two cases to consider.We have namely that T = 0 and cos(θ) = 0. In the first case we obtainthat the hypersurface M is everywhere orthogonal to ∂n+2. Hence weobtain the first family hypersurfaces. In the second case we obtain thatM is an open part of Mn−1 × R, where Mn−1 is a hypersurface of Hn.One can easily show that Mn−1 × R is a totally geodesic hypersurfaceof Hn × R if and only if Mn−1 is a totally geodesic hypersurface of Hn.


We obtain in this way the second family of hypersurfaces stated in thetheorem.

The following proposition is the counterpart of Proposition 1 in [58], andits proof is similar.

Proposition 4.4.3. Let M be a totally umbilical hypersurface of Hn×Rwith angle function θ. Then there exist local coordinates (u, v1, . . . , vn−1)on M , such that θ only depends on u and such that the following equationholds:

φ′′ − sinφ = 0, with φ = 2θ. (4.14)

Conversely, starting with an open subset U ⊆ Rn with coordinates(u, v1, . . . , vn−1) and a solution φ of (4.14), we can define a functionλ and a Riemannian metric on U such that there exist an isometricimmersion F : U → Hn × R with shape operator S = λId and anglefunction θ = φ

2 .

We can now prove the classification of totally umbilical hypersurfaces ofHn × R.

Theorem 4.4.4. Let M be a totally umbilical hypersurface of Hn × R,which is not totally geodesic, with angle function θ. Then there exists alocal coordinate system (u, v1, . . . , vn−1) on M such that θ only dependson u, the shape operator is S = θ′Id and

(θ′)2 − sin2 θ = c, (4.15)

where c > −1 is a real constant. Moreover, M is locally congruent to arotation hypersurface of Hn × R for which the profile curve α and theaxis P 2 are given by

• α(u) = 1√c

(θ′, 0, . . . , sin θ,

√c

∫sin θdu

), P 2 = span{∂1, ∂n+2},

if c > 0,

• α(u) = 1√−c

(sin θ, 0, . . . , θ′,

√−c∫

sin θdu), P 2 = span{∂n+1,

∂n+2}, if 0 > c > −1,

• α(u) =(θ′, 0, . . . , 0,− 1

2θ′ ,∫

sin θdu)

with respect to the pseudo-

orthonormal basis {e1, . . . , en+2}, P 2 = span{en+1, en+2}, if c = 0.


Conversely, all rotation hypersurfaces with profile curves and axes givenabove, where θ and c satisfy (4.15), are totally umbilical in Hn × R.

Proof. Let M be a totally umbilical hypersurface of Hn×R, which is nottotally geodesic, with shape operator S = λId and angle function θ. Itfollows from Proposition 4.4.3 that there exist a local coordinate system(u, v1, . . . , vn−1) on U ⊂M such that λ and θ depend only on u and that(θ′)2 − sin2 θ = c is a constant. Remark that c ≥ −1 and that c = −1 ifand only if cos θ = 0 everywhere, i.e. M is an open part ofMn−1×R withM

n−1 a hypersurface of Hn. It is easily verified thatMn−1×R is a totallyumbilical hypersurface of Hn × R if and only if it is a totally geodesichypersurface. Suppose θ is a function depending on u and satisfyingequation (4.15). Define a Riemannian metric g = du2 + sin2 θ gc on U ,where gc is a Riemannian metric of constant sectional curvature c. Wewill use Theorem 2.1.2 to show that there, up to isometries of Hn × R,exist a unique immersion F : U → Hn × R, such that F is isometric, theprojection of ∂n+2 on F (U) is F∗(sin θ∂u), the angle between the unitnormal N and ∂n+2 is θ and the shape operator S = θ′Id. We have toconsider three cases depending on whether c > 0, 0 > c > −1 or c = 0.For the first case we consider the immersion

F (u, v1, . . . , vn−1) = 1√c(θ′, sin θφ1(u, v1, . . . ,

vn−1), . . . , sin θφn(u, v1, . . . , vn−1),√c

∫ u

u0sin θ(σ)dσ),

where (φ1(u, v1, . . . , vn−1), . . . , φn(u, v1, . . . , vn−1)) is a parametrizationof Sn−1 in En. It is easily verified by a straightforward calculation thatF satisfies the necessary conditions. Remark that F is a parametrizationof a rotation hypersurface of Hn × R about P 2 = span{∂1, ∂n+2} withprofile curve of the first case of the theorem. For the second case weconsider the immersion

F (u, v1, . . . , vn−1) = 1√−c

(sin θφ1(u, v1, . . . , vn−1), . . . ,

sin θφn(u, v1, . . . , vn−1), θ′,√−c∫ u

u0sin θ(σ)dσ),


where (φ1(u, v1, . . . , vn−1), . . . , φn(u, v1, . . . , vn−1)) is now a parametriza-tion of Hn−1 in Ln. A straightforward computation yields that thisimmersion satisfies the necessary conditions. In this case we have thatF is a parametrization of a rotation hypersurface of Hn × R aboutP 2 = span{∂n+1, ∂n+2} with profile curve of the second case of thetheorem. Finally, we will consider the last case, where c = 0. Considertherefore the immersion

F (u, v1, . . . , vn−1) = (θ′, θ′v1, . . . , θ′vn−1,

− 12θ′ −

θ′

2 (n−1∑i=1

v2i ),∫ u

u0sin θ(σ)dσ),

with respect to the pseudo-orthonormal basis {e1, . . . , en+2}. It followsby a straightforward calculation that F satisfies the necessary conditions.Notice that F is a parametrization of the rotation hypersurface of thethird case of the theorem.

4.4.4 Semi-parallel hypersurfaces of Hn × R

In this section we will classify the semi-parallel hypersurfaces of Hn × R.Again, if we compare our result to the classification in Sn×R, given in [58],some new interesting families arise. The following lemma characterizesthe semi-parallel hypersurfaces in terms of the shape operators and canbe proven in the same way as its spherical counterpart in [58].

Lemma 4.4.5. Let M be a semi-parallel hypersurface of Hn×R. DenoteT and θ as above. Then there exists a local orthonormal frame field{e1, . . . , en} on M such that the shape operator takes one of the followingforms with respect to the orthonormal frame:

1. S = λId

2. S =

λ

µµ

. . .µ

with λµ = cos2 θ and if n ≥ 3, then

T = ‖T‖e1,


3. S =

0λ

. . .λ

µ. . .

µ

with λµ = 1 and e1 = T =

∂n+2.

Using Lemma 4.4.5, we obtain the classification of the semi-parallelhypersurfaces in Hn × R.

Theorem 4.4.6. Let M be a semi-parallel hypersurface of Hn×R. Thenthere are four possibilities:

1. n = 2 and M is flat,

2. M is totally umbilical,

3. M is an open part of a rotation hypersurface for which the profilecurve can be parameterized as

• α(s) =(

cosh(s), 0, . . . , 0, sinh(s),±∫ s

s0

√C cosh2(σ)− 1 dσ

)if P 2 = span{∂1, ∂n+2} is Lorentzian,

• α(s) =(

cosh(s), 0, . . . , 0, sinh(s),±∫ s

s0

√C sinh2(σ)− 1 dσ

)if P 2 = span{∂n+1, ∂n+2} is Riemannian,

• α(s) =(s, 0, . . . ,− 1

2s,±∫ s

s0

√C − 1

2σ2 dσ

), with respect to

{e1, . . . , en+2} if P 2 = span{en+1, en+2} is degenerate.

4. M ⊆Mn−1×R where Mn−1 is a semi-parallel hypersurface of Hn.

Proof. Suppose that M is a semi-parallel hypersurface of Hn × R withshape operator S. Lemma 4.4.5 yields that there are three possible formsof S to consider.

In the first case of Lemma 4.4.5, M is a totally umbilical hypersurface.This gives us the second case of the theorem.


If we are in the second case of Lemma 4.4.5 and n = 2, then M2

is a general flat surface H2 × R. This gives us the first case of thetheorem. If n ≥ 3, then the form of S is the one given in Theorem 4.4.1,characterizing rotation hypersurfaces. We also have that λµ = cos2 θ.This is not a functional relation in the strict sense, because θ can bea non-constant function. However, we see from (4.7) that θ does notvary in directions orthogonal to T . By the proof of Theorem 2 in [20],this is actually enough to conclude that M is a rotation hypersurface ofHn × R. We have to consider now three cases, depending on whetherthe rotation axis P 2 is Riemannian, Lorentzian or degenerate. Assumefirst that P 2 is Riemannian and spanned by ∂n+1 and ∂n+2. From theequation λµ = cos2 θ, we will be able to determine the profile curve ofthe rotation hypersurface M . Remark that this equation is satisfied inthe case that the profile curve is a vertical line. We are then in case 4of the theorem, where Mn−1 is a totally umbilical hypersurface of Hn.If the profile curve is not a vertical line, it can be parameterized asα(s) = (cosh(s), 0, . . . , 0, sinh(s), a(s)) and the formulae (4.9) give that

λµ = a′(s)a′′(s) tanh(s)1 + a′(s)2 .

We also have that

cos2 θ = 1−sin2 θ = 1−〈∂n+2,T

‖T‖〉2 = 1−〈∂n+2,

α′

‖α′‖〉2 = 1

(1 + a′(s)2)2 .

Hence, the equation λµ = cos2 θ becomes a′′(s)a′(s) tanh(s) = 1 + a′(s)2,or equivalently a′(s)2 = C sinh2(s)−1 with C a real constant. This coversthe second subcase of case 3 of the theorem. Suppose now that the axis ofrotation P 2 is Lorentzian and spanned by ∂1 and ∂n+2. We will determinethe profile curve by using equation cos2 θ = λµ. This equation is certainlysatisfied for vertical lines. We are then in case 4 of the theorem, whereM

n−1 is a totally umbilical hypersurface of Hn. Hence, we can supposethat the profile curve is not a vertical line and can be parameterizedas α(s) = (cosh(s), 0, . . . , 0, sinh(s), a(s)). Using equations (4.8), we seethat cos2 θ = λµ is equivalent to 1+a′(s)2 = a′′(s)a′(s) coth(s). It can beeasily verified that this equation is equivalent to a′(s)2 = C cosh(s)2 − 1,with C ∈ R and so we obtain the first subcase of case 3 of the theorem.Finally, we suppose that the rotation axis P 2 is degenerated and spannedby en+1 and en+2. It is clear that equation cos2 θ = λµ is satisfied if the


profile curve is a vertical line. We are then in case 4 of the theorem,where Mn−1 is a totally umbilical hypersurface of Hn. Suppose now thatthe profile curve is not a vertical line and hence can be parameterizedas α(s) = (s, 0, . . . , 0,− 1

2s , a(s)) with respect to the pseudo-orthonormalbasis {e1, . . . , en+1, en+2}. The formulas (4.12) give that

λµ = sa′(s)(sa′(s) + s2a′′(s))(1 + s2a′(s)2)2 .

We also have thatcos2 θ = 1

1 + s2a′(s)2 .

Hence we have that equation cos2 θ = λµ is equivalent to 1 + s2a′(s)2 =s2a′(s)2 +s3a′(s)a′′(s), for which the general solution is given by a′(s)2 =C − 1

2s2 , with C ∈ R. So we deduced the third subcase of case 3 ofthe theorem. In the last case of Lemma 4.4.3, we have that ∂n+2 is

tangent to M and hence M is an open part of Mn−1 × R, where Mn−1

is a hypersurface of Hn. Since Hn is a totally geodesic hypersurfaceof Hn × R, we have that the shape operator S of Mn−1 in Hn satisfiesSX = SX for X tangent to Mn−1. It follows that S takes the form

S =

λ. . .

λµ

. . .µ

,

with λµ = 1. It was proven in [19] that this is the shape operator of asemi-parallel hypersurface of Hn.

4.4.5 Parallel hypersurfaces in Hn × R

It is well known that a parallel hypersurface is also semi-parallel. In factthe condition of semi-parallelism is the integrability condition for thesystem of differential equations in the components of h that expresses thecondition of parallelism. However, in the case of Hn×R the classificationof parallel hypersurfaces follows directly from the Codazzi equation.

EXTRINSIC SYMMETRIES IN M(C1)×M(C2) 81

Theorem 4.4.7. Let M be a parallel hypersurface of Hn × R. Thenthere are two possibilities:

1. M is an open part of a totally geodesic hypersurface Hn × {t0},

2. M is an open part of a Riemannian product Mn−1 × R, whereM

n−1 is a parallel hypersurface of Hn.

Proof. If M is a parallel hypersurface of Hn×R, then it follows from theCodazzi equation (4.5) that at any point of the hypersurface either T = 0or cos θ = 0. Since T and cos θ are continuous and ‖T‖2 = 1−cos2 θ, oneof these must hold on the whole hypersurface. In the first case, ∂n+2 iseverywhere orthogonal toM . Hence, M is an open part of a hypersurfaceof type Hn × {t0}. In the second case, we have that ∂n+2 is everywheretangent to M and therefore M is an open part of a hypersurface oftype Mn−1 × R, with Mn−1 a hypersurface of Hn. It is easy to see thatM

n−1×R is parallel in Hn×R if and only if Mn−1 is parallel in Hn.

4.5 Extrinsic symmetries in M(c1)×M(c2)

In this section we will study totally umbilical, semi-parallel and parallelhypersurfaces M in M(c1) ×M(c2), where M(c1) and M(c2) are realspace forms such that dimM(c1) = n1 ≥ 2, dimM(c2) = n2 ≥ 2, c1c2 6= 0and dimM(c1) + dimM(c2) = n + 1. We recall first all the importantdefinitions and notations.

4.5.1 Hypersurfaces of M(c1)×M(c2)

In section 2.3 we have seen that M(c1) ×M(c2) can be isometricallyimmersed in En+3,Ln+3 and Rn+3

2 if c1, c2 > 0, c1c2 < 0 and c1, c2 < 0,respectively. The Levi-Civita connection ∇ and the associated Riemanncurvature tensor R are given by equations (2.35) and (2.36) in terms ofthe Levi-Civita connection D of En+3,Ln+3 or Rn+3

2 and the productstructure F ofM(c1)×M(c2). Let f, u, U and λ be a linear endomorphismof the tangent bundle, a 1-form, a vector field and a function onM , respectively, defined by equations (2.24) and (2.25). Moreoverwe have that f, u, U and λ satisfy equations (2.26) − (2.30). TheGauss and Codazzi equation are now given in terms of f and U by


equations (2.49) and (2.50). The covariant derivatives of f, u, U and λare given by equations (2.51), (2.52) and (2.53). These equations appearin the existence and uniqueness theorem for isometric immersions ofhypersurfaces of M(c1) ×M(c2), see Theorem 2.3.5. We can define afunction θ : M → [0, π2 ] such cos(2θ) = λ, because ‖U‖2 + λ2 = 1. Wecall θ the angle function of the hypersurface M of M(c1)×M(c2).

4.5.2 Totally umbilical hypersurfaces

In this section we will consider totally umbilical hypersurfaces ofM(c1)×M(c2). But first we will proof that there are not so many totally geodesichypersurfaces in M(c1)×M(c2).

Theorem 4.5.1. Let M be a totally geodesic hypersurface in M(c1)×M(c2). Then M is an open part of M ×M(c2) or M(c1) ×M , whereM and M are totally geodesic hypersurfaces of M(c1) and M(c2),respectively.

Proof. Let M be a totally geodesic hypersurface in M(c1)×M(c2). Weobtain from equation of Codazzi (2.50) that

c1I + f

2 ((X ∧ Y )U)− c2I − f

2 ((X ∧ Y )U) = 0. (4.16)

Suppose first that U = 0. Then we have that equation (4.16) is triviallysatisfied and one can also easily deduce that M is an open part ofM ×M(c2) or an open part of M(c1)×M . Moreover it is easy to seethat M ×M(c2) and M(c1)×M are totally geodesic hypersurfaces inM(c1)×M(c2) if and only if M andM are totally geodesic hypersurfacesof M(c1) and M(c2), respectively.

Suppose that there exists a point in M , where U 6= 0. We can findthen an open neighbourhood O of p in M , where U 6= 0. Denote byU⊥ the orthogonal complement of U in TO. it can be easily shown thatU⊥ = D1 ⊕ D−1, where D1 = {X ∈ TO|fX = X} and D−1 = {X ∈TO|fX = −X}. Moreover we have that dimD1 = dimM(c1) − 1 ≥ 1and dimD−1 = dimM(c2)− 1 ≥ 1. Take now X ∈ D1 and Y = U , thenusing the equation of Codazzi we obtain that

c1〈U,U〉X = 0.


Because c1 6= 0, we obtain that 〈U,U〉 = 0 and we have a contradiction.So we obtain the family of hypersurfaces stated in the theorem.

Proposition 4.5.2. Let M be a totally umbilical hypersurface inM(c1) ×M(c2) with angle function θ and let p be a point in M suchthat θ(p) /∈ {0, π2 }. Then c1 + c2 = 0 and there exist local coordinates(∂u, ∂v1

, . . . , ∂vn1−1, ∂v1 , . . . , ∂vn2−1) on a open neighborhood O of p in M

such that θ depends only on u and such that φ := 2θ satisfies the followingequation

φ′′ = c1 sin(φ). (4.17)Conversely, starting with an open subset O ⊆ Rn with coordinates(u, v1, . . . , vn1−1, v1, . . . , vn2−1) and a solution φ(u) of φ′′ = c1 sin(φ),which is nowhere zero on O, we can put θ = φ

2 and we can define afunction µ and a Riemannian metric on O such that there exists anisometric immersion ψ : O → M(c1) × M(−c1) with shape operatorS = µI and angle function θ.

Proof. Suppose that M is an totally umbilical hypersurface with shapeoperator S = µI. Since U 6= 0 at the point p, we can choose an openneighbourhood O of p in M , where U 6= 0. Let X,Y be vector fieldstangent to O. Then the equations (2.51) and (2.53) yield

(∇Xf)Y = µ(u(Y )X + 〈X,Y 〉U), (4.18)

Xλ = −2µu(X). (4.19)

Take now X,Y ∈ D1 = {X ∈ TO|fX = X}. Using equation (4.18), wecan easily show that [X,Y ] ∈ D1. We have namely that

0 = (∇Xf)Y −(∇Y f)X = ∇XfY −∇Y fX−f [X,Y ] = [X,Y ]−f [X,Y ].

Analogously one can prove that [X,Y ] ∈ D−1 if X,Y ∈ D−1 = {X ∈TO|fX = −X}. We also have that ∇XY ∈ D−1 and ∇YX ∈ D1 ifX ∈ D1 and Y ∈ D−1, because using equation (4.18) we obtain that

∇XfY − f∇XY = ∇Y fX − f(∇YX) = 0.

Hence we can choose a coordinates (u, v1, . . . , vn1−1, v1, . . . , vn2−1) on Osuch that ∂vi ∈ D1, ∂vj ∈ D−1 and ∂u = U

‖U‖ . From equation (4.19) weobtain that ∂viλ = ∂vjλ = 0 and

θ′ = µ. (4.20)


The equation of Codazzi yields

Y µX −XµY = c1I + f

2

(12(X ∧ Y )U

)− c2

I − f2

(12(X ∧ Y )U

).

Suppose that X,Y ∈ U⊥ and X ⊥ Y , then we obtain from the previousequation that Xµ = Y µ = 0. Suppose now that X ∈ D1 and Y = U ,then we obtain form the equation of Codazzi that

Uµ = c1〈U,U〉

2 . (4.21)

Analogously we obtain that

Uµ = −c2〈U,U〉

2 , (4.22)

if X ∈ D−1 and Y = U . We deduce form the previous equations thatc1 + c2 = 0, because ‖U‖ 6= 0, and that

µ′ = c1 cos(θ) sin(θ). (4.23)

Remark that from equation (4.20) and (4.23), we obtain that θ satisfiesθ′′ = c1 sin(θ) cos(θ).

Conversely, let us start with an open neighbourhood O ⊆ Rn, withcoordinates (u, v1, . . . , vn1−1, v1, . . . , vn2−1) and with a non-vanishingsolution of (4.17). We define a metric g on U by

du2 +n1−1∑i,j=1

gijdvidvj +n2−1∑α,β=1

gαβdvαdvβ + 2n1−1∑i=1

n2−1∑α=1

giαdvidvα,

the function θ(u) by φ(u)2 , the vector field U by sin(2θ)∂u, the function λ

by cos(2θ), the field of operators S by θ′I and the field of operators f byf∂vi = ∂vi , f∂vα = −∂vα and f∂u = − cos(2θ)∂u. By direct calculationand definition of the data, one can show that this data satisfies thelast three equations of (2.48), the Codazzi equation (2.50) and equation(2.53). It is also easy to see that the data satisfies the first equation of(2.48) if and only if giα = 0. So in the following we suppose that ourmetric g is given by

du2 +n1−1∑i,j=1

gijdvidvj +n2−1∑α,β=1

gαβdvαdvβ.


From Theorem 2.3.5, we know that there exists an isometric immersion of(O, g) into M(c)×M(−c) with shape operator S, vector field U , functionλ and field of endomorphism f if and only if the equation of Gauss (2.49)and equations (2.51) and (2.52) are satisfied. By direct calculation onecan show that these equations are equivalent to

R(∂vi , ∂u)∂u = ((θ′)2 + c sin2(θ))∂vi , (4.24)

R(∂vα , ∂u)∂u = ((θ′)2 − c cos2(θ))∂vα , (4.25)

R(∂u, ∂vi)∂vα = R(∂vi , ∂vα)∂u = R(∂u, ∂vα)∂vi = 0, (4.26)

R(∂vi , ∂vj )∂u = R(∂vα , ∂vβ )∂u = 0, (4.27)

R(∂vi , ∂vj )∂vα = R(∂vα , ∂vβ )∂vi = 0, (4.28)

R(∂vα , ∂vi)∂vj = g(∂vi , ∂vj )(θ′)2∂vα , (4.29)

R(∂vi , ∂vα)∂vβ = g(∂vα , ∂vβ )(θ′)2∂vi , (4.30)

R(∂vi , ∂vj )∂vk = ((θ′)2 + c)(∂vi ∧ ∂vj )∂vk , (4.31)

R(∂vα , ∂vβ )∂vγ = ((θ′)2 − c)(∂vα ∧ ∂vβ )∂vγ , (4.32)

∇∂vi

∂vj − f∇∂vi∂vj = θ′ sin(2θ)gij∂u, (4.33)

−∇∂vα∂vβ − f∇∂vα∂vβ = θ′ sin(2θ)gαβ∂u, (4.34)

f∇∂vi

∂vα = −∇∂vi

∂vα , (4.35)

f∇∂vα∂vi = ∇∂vα∂vi , (4.36)

∇ is the Levi-Civita connection of (O, g) and R is the Riemann curvaturetensor associated to ∇. From the last four equation we deduce that

∂ugij = ∂u〈∂vi , ∂vj 〉 = 〈∇∂u∂vi , ∂vj 〉+ 〈∂vi ,∇∂u∂vj 〉

= −2θ′ tan(θ)gij ,

∂ugαβ = 2θ′ cot(θ)gαβ,

∂vigαβ = 〈∇∂vi

∂vα , ∂vβ 〉+ 〈∂vα ,∇∂vi

∂vβ 〉 = 0,


∂vαgij = 0.

Hence we obtain that

gij = cos2(θ)cij(v1, . . . , vn1−1),

gαβ = sin2(θ)cαβ(v1, . . . , vn2−1).

and that the metric g = du2 + cos2(θ)gc + sin2(θ)gc is a warped productmetric with warping functions cos(θ) and sin(θ). Now there rise thequestion if we can find metrics gc and gc such that equations (4.24)−(4.32)are satisfied. From Proposition 4.3.1 we immediately see that equations(4.26), (4.27) and (4.28) are satisfied for any choice of cij and cαβ . UsingProposition 4.3.1, we show that equations (4.29) and (4.30) are satisfiedfor any choice of cij and cαβ. Using Proposition 4.3.1 we calculateR(∂vα , ∂vi)∂vj

R(∂vα , ∂vi)∂vj = −g(∂vi , ∂vj )g(grad cos(θ), grad(sin(θ)))

cos(θ) sin(θ) ∂vα

= g(∂vi , ∂vj )(θ′)2∂vα .

(4.37)

Hence equation (4.29) is satisfied. Analogously one can show thatequation (4.30) is satisfied. Next we show that equations (4.24) and(4.25) are satisfied. We calculate therefore R(∂vi , ∂u)∂u. We obtain usingProposition 4.3.1 and equation (4.17)

R(∂vi , ∂u)∂u = −Hcos(θ)(∂u, ∂u)

cos(θ) ∂vi

= θ′′ sin(θ)− (θ′)2 cos(θ)cos(θ) ∂vi

= (c sin2(θ) + (θ′)2)∂vi .

(4.38)

Analogously one can show that equation (4.25) is satisfied. We checknow if equations (4.31) and (4.32) are satisfied. Like previously we willuse Proposition 4.3.1 to check these equations.

R(∂vi , ∂vj )∂vk = Rc(∂vi , ∂vj )∂vk −‖grad(cos(θ))‖2

cos2(θ) ((∂vi ∧ ∂vj )∂vk)

= Rc(∂vi , ∂vj )∂vk − tan2 θ(θ′)2(∂vi ∧ ∂vj )∂vk ,


where Rc is the Riemann curvature tensor associated to the metric gc.We obtain that equation (4.24) is equivalent to

Rc(∂vi , ∂vj )∂vk = ((θ′)2 + c cos2(θ))(∂vi ∧gc ∂vj )∂vk . (4.39)

Analogously we obtain that equation (4.25) is equivalent to

Rc(∂vα , ∂vβ )∂vγ = ((θ′)2 − c sin2(θ))(∂vα ∧gc ∂vβ )∂vγ . (4.40)

We notice that (θ′)2 + c cos2(θ) and (θ′)2− c sin2(θ) are constant, becauseθ satisfies equation θ′′ = c cos(θ) sin(θ). Hence we have that the equations(4.31) and (4.32) are satisfied if and only if we choose the metrics gcand gc such that the metrics gc and gc have constant constant sectionalcurvature c and c, respectively, such that c− c = c.

We are now ready to classify all the totally umbilical hypersurfaces ofM(c1)×M(c2) with c1c2 6= 0. We have already seen that we necessarilyhave that c1 + c2 = 0. We suppose now that M(c1)×M(c2) = M(c)×M(−c) with c > 0. Moreover we can suppose that c = 1. This will makethe calculations easier.

Theorem 4.5.3. Let M be a totally umbilical hypersurface of Sn1 ×Hn2, which is not totally geodesic, with angle function θ such λ =cos(2θ) and let p be a point of M where cos(θ) sin(θ) 6= 0. Then thereexists a local coordinate system (u, v1, . . . , vn1−1, v1, . . . , vn2−1) on a openneighbourhood O of p in M such that θ only depends on u, the shapeoperator is S = θ′I and

(θ′)2 + cos2(θ) = c, (4.41)

where c is a strictly positive real number. Moreover,M is locally isometricto

1. c > 1: φ : Rn → Sn1 × Hn2(u, v1, . . . , vn1−1, v1, . . . , vn2−1) →( θ′√

c, cos(θ)√

cφ, θ

′√c, sin(θ)

c φ1),

2. 0 < c < 1: φ : Rn → Sn1 ×Hn2 : (u, v1, . . . , vn1−1, v1, . . . , vn2−1)→( θ′c, cos(θ)

cφ, sin(θ)√

−c φ2,θ′√−c),

3. c = 1: φ : Rn → Sn1 × Hn2 : (u, v1, . . . , vn1−1, v1, . . . , vn2−1) →(θ′, cos(θ)φ, θ′, v1θ

′, . . . , vn2−1θ′,− 1

2θ′ −θ′

2∑α v

2α),


where c − c = 1 and φ = φ(v1, . . . , vn1−1), φ1 = φ1(v1, . . . , vn2−1) andφ2 = φ2(v1, . . . , vn2−1) are orthogonal parametrizations of Sn1−1, Sn2−1

and Hn2−1, respectively. Conversely, all the hypersurfaces stated above,where θ, c and c satisfy equation (4.41) and c = c−1, are totally umbilicalhypersurfaces of Sn1 ×Hn2.

Proof. Suppose we have a totally umbilical hypersurface in Sn1 ×Hn2−1,which is not totally geodesic, with shape operator S = µI andangle function θ. From Proposition 4.5.2 it follows that there existsa local coordinate system (u, v1, . . . , vn1−1, v1, . . . , vn2−1) on a openneighbourhood O of p in M such that µ and θ only depend on u andthat (θ′)2 + cos2(θ) = c. Let us remark that c ≥ 0 and c = 0 ifand only is cos(θ) = 0 everywhere, i.e. M is an open part Sn1 ×Mwith M a hypersurface of Hn2 . One can easily verify that Sn1 ×M isa totally umbilical hypersurface if and only if it is a totally geodesichypersurface of Sn1 × Hn2 . Suppose θ is a function depending onlyon u and satisfying equation (4.41). Define a Riemannian metricg = du2 + cos2(θ)gc + sin2(θ)gc, where gc and gc are Riemannian metricsof constant sectional curvature c and c, respectively. We will use Theorem2.3.5 to show that there, up to isometries of Sn1 ×Hn2 , exist a uniqueimmersion φ, such that φ is isometric, Fφ∗(X) = φ∗(fX) + g(X,U)Nand FN = φ∗(U) + cos(2θ)N and the shape operator S = θ′I,where U = sin(2θ)∂u and f is a field of endomorphisms such thatf∂vi = ∂vi , f∂vα = −∂vα and f∂u = − cos(2θ)∂u. We remark that wehave to consider three cases depending on whether c > 1, 1 > c > 0 andc = 1. For the first case we consider the immersion

φ(u, v1, . . . , vn1−1, v1, . . . , vn2−1) = ( θ′√c,cos(θ)√

cφ,

θ′√c,sin(θ)c

φ1),

where c = c − 1 and φ = φ(v1, . . . , vn1−1) and φ1 = φ1(v1, . . . , vn2−1)are orthogonal parameterizations of Sn1−1 and Sn2−1 in En1 and En2 ,respectively. It is easily verified by a straightforward calculation that φsatisfies the necessary conditions. Analogously we obtain that the othertwo immersions are totally umbilical hypersurfaces in Sn1 × Hn2 . Weremark here that the last parametrization is written with respect to apseudo-orthonormal basis.


4.5.3 Semi-parallel hypersurfaces

In this section we will classify semi-parallel hypersurfaces in M(c1) ×M(c2). The following lemma will give us the shape operator of semi-parallel hypersurfaces in M(c1)×M(c2).

Proposition 4.5.4. Let M be a semi-parallel hypersurface in M(c1)×M(c2), with c1c2 6= 0 and dimM(ci) = ni ≥ 2 for i = 1, 2, and the vectorfield U and function λ = cos(2θ) are defined as before. Then there existsa local orthornormal frame {e1, . . . , en} such that the shape operator Sof M takes the following form with respect to the orthonormal frame{e1, . . . , en}:

1.

µ. . .

µ

,

2.

S 0

0

0 . . .0

, with sin(θ) = 0 and where S has the form

of a shape operator of a semi-parallel hypersurface in M(c1),

3.

0. . . 0

0

0 S

, with cos(θ) = 0 and where S has the form

of a shape operator of a semi-parallel hypersurface in M(c2),

Proof. Suppose we have a hypersurface M in M(c1) × M(c2), withc1c2 6= 0 and dimM(c1) = n1 ≥, dimM(c1) = n2 ≥ 2. Denote by Sthe shape operator of M in M(c1) ×M(c2) and let {e1, . . . , en} be anorthonormal tangent frame of M such that Sei = λiei. Let us firstassume that ‖U‖2 = sin2(2θ) = 0. We have then that M is an open partof M ×M(c2) if sin(θ) = 0 or an open part of M(c1)×M if cos(θ) = 0,


where M and M are hypersurfaces of M(c1) and M(c2), respectively. Itis easy to see that the shape operator S is given by

S 0

0

0 . . .0

or

0. . . 0

0

0 S

,

where S and S are the shape operators of M and M , respectively. Itis easy to see, using the equation of Gauss, that M is a semi-parallelhypersurface if and only if the shape operator is given by 2 or 3.

Let us now assume that U 6= 0. We have then that the distributionsD1 = {X ∈ TM | fX = X} and D−1 = {X ∈ TM | fX = −X} haveconstant dimensions n1−1 and n2−1, respectively, and fU = − cos(2θ)U .Using the equation of Gauss, we obtain

R(ei, ej)X = λiλj(ei ∧ ej)X + c1P (ei ∧ ej)X,

R(ei, ej)X = λiλj(ei ∧ ej)X + c2Q(ei ∧ ej)X,

R(ei, ej)U = λiλj(ei ∧ ej)U

+c1 sin2(θ)P (ei ∧ ej)U + c2 cos2(θ)Q(ei ∧ ej)U,

where X ∈ D1, X ∈ D−1, P = I+f2 and Q = I−f

2 and i and j are different.From these equations, we obtain that M is a semi-parallel hypersurfaceif and only if

(λj − λi)(λiλj(ei ∧ ej)X + c1P (ei ∧ ej)X) = 0, (4.42)

(λj − λi)(λiλj(ei ∧ ej)X + c2Q(ei ∧ ej)X) = 0, (4.43)

(λj − λi)(λiλj(ei ∧ ej)U

+c1 sin2(θ)P (ei ∧ ej)U + c2 cos2(θ)Q(ei ∧ ej)U) = 0. (4.44)

Suppose first that all the eigenvalues of S are equal, then we are incase 1 of the lemma. From now on we suppose that S has at least


two different eigenvalues λi and λj . Let us fix the two indices i 6= j.We consider two cases, namely (ei ∧ ej)U = 0 and (ei ∧ ej)U 6= 0.Suppose we are in the first case, then we have that equation (4.44) issatisfied. We have now three subcases to consider, namely (ei ∧ ej)X = 0for every X ∈ D1, (ei ∧ ej)X = 0 for every X ∈ D−1 or there existX ∈ D1 and X ∈ D−1 such that (ei ∧ ej)X 6= 0 and (ei ∧ ej)X 6= 0.Let us consider the first subcase then we obtain using the fact that fis symmetric and the definitions of U,D1 and D−1 that {ei, ej} ⊂ D−1and λiλj + c2 = 0. Analogously we obtain that {ei, ej} ⊂ D1 andλiλj + c1 = 0 when we consider the second subcase. Finally let usconsider the third subcase, we obtain from equations (4.42) and (4.43)that (ei∧ej)X and (ei∧ej)X are eigenvectors of P and Q with eigenvalues−λiλj

c1and −λiλj

c2, respectively. Let us remark that (ej ∧ ej)X /∈ sp{U}

and (ej ∧ ej)X /∈ sp{U}, because (ei ∧ ej)U = 0. Hence we obtainthat (ei ∧ ej)X and (ei ∧ ej)X lie in D1 or D−1. We obtain now twocases, we either have that (ei ∧ ej)X ∈ D1 and (ei ∧ ej)X ∈ D−1 withλiλj + c1 = λiλj + c2 = 0 or that (ei ∧ ej)X ∈ D−1 and (ei ∧ ej)X ∈ D1with λiλj = 0. We show now that first cases can not occur. Fromequation (4.42) we obtain that (ei ∧ ej)2X ∈ D1 if (ei ∧ ej)X ∈ D1.Since (ei ∧ ej)X 6= 0 we have that {ei, ej} ⊂ D1. Analogously we havethat {ei, ej} ⊂ D−1. Since D1 and D−1 are orthogonal, we obtain acontradiction and this case can not occur.

Suppose now we are in the second case for which we have that (ei∧ej)U 6=0. Remark that (ei ∧ ej)U ∈ D1 ⊕ D−1. From equation (4.44) wededuce that (ei ∧ ej)U is an eigenvector of c1 sin2(θ)P + c2 cos2(θ)Qwith eigenvalue −λiλj . We suppose first that c1 sin2(θ) 6= c2 cos2(θ).Hence we have that (ei ∧ ej)U ∈ D1 and λiλj + c1 sin2(θ) = 0 or that(ei ∧ ej)U ∈ D−1 and λiλj + c2 cos2(θ) = 0. Suppose we are in the firstsubcase. From equation (4.42) and the fact that λiλj + c1 sin2(θ) = 0, weobtain that (ei∧ej)X ∈ sp{U}. From the previous one can easily deducethat U ∈ sp{ei, ej} and that {ei, ej} ∈ D1 ⊕ sp{U}. Analogously wehave that λiλj + c2 cos2(θ) = 0, U ∈ sp{ei, ej}, {ei, ej} ⊂ D−1 ⊕ sp{U}and (ej ∧ ej)X ∈ sp{U} for all X ∈ D−1. Suppose now that c1 sin2(θ) =c2 cos2(θ), then we have that λiλj + c1 sin2(θ) = λiλj + c2 cos2(θ) = 0.We therefore have that cos(2θ) is constant and that c1c2 > 0. We showthat we have a contradiction in this case. From equations (4.42) and(4.43) we obtain that (ei ∧ ej)X ∈ sp{U} and (ei ∧ ej)X ∈ sp{U} forall X ∈ D1 and for all X ∈ D−1. Hence we obtain that U ∈ sp{ei, ej},


because (ei ∧ ej)U 6= 0, (ei ∧ ej)X ∈ sp{U} for all X ∈ D1 ⊕ D−1. Wehave that cos(2θ) is a constant and hence from equation (2.53) we obtainthat λig(U, ei)2 + λjg(U, ej)2 = g(SU,U) = 0. So we have that λi = 0or λj = 0. This gives us a contradiction, because form the previous wehave that sin(θ) = cos(θ) = 0. We conclude from previous calculations:if λi and λj are different eigenvalues of S, then

λiλj + c1 = 0 and {ei, ej} ⊂ D1, (4.45)

λiλj + c2 = 0 and {ei, ej} ⊂ D−1, (4.46)

λiλj + c1 sin2(θ) = 0, (ei ∧ ej)U ∈ D1 and (ei ∧ ej)D1 ⊂ sp{U}, (4.47)

λiλj + c2 cos2(θ) = 0, (ei ∧ ej)U ∈ D−1 and (ei ∧ ej)D−1 ⊂ sp{U},(4.48)

λiλj = 0, (ei ∧ ej)U = 0, (ei ∧ ej)D1 ⊂ D−1 and (ei ∧ ej)D−1 ⊂ D1.(4.49)

Let us assume now that S has exactly two different eigenvalues λ andµ. So we have that Sei = λei for i ∈ {1, . . . , k} and that Sej = µejfor j ∈ {k + 1, . . . , n}. If λµ + c1 = 0, λµ + c2 = 0 or λµ = 0, thenwe have that U is perpendicular to sp{ei, ej} for every i ∈ {1, . . . , k}and for every j ∈ {k + 1, . . . , n}. Hence we obtain that U = 0 and thisgives us a contradiction. Suppose now that λµ+ c1 sin2(θ) = 0 or thatλµ+ c2 cos2(θ) = 0, then we have that k = 1 or that n = k− 1. It is easyto see that then we have that dimD1 = 0 or that dimD−1 = 0 and hencen1 = 1 or n2 = 1. This is a contradiction, because we have assumed thatn1, n2 > 1.

Now let us assume that S has at least three different eigenvalues λ, νand µ. If λµ + c1 = λν + c1 = 0, then we have µ = ν, which is acontradiction. If λµ + c1 = λν + c2 = 0, then the eigenvector eλ of Swith eigenvalue λ lies in D1 and D−1, this is of course a contradiction.If λµ + c1 = 0 = λν, then the eigenvector eν of S with eigenvalue νlies in D−1. If λµ + c1 = λν + c1 sin2(θ) = 0, then eν lies in sp{U}.Moreover we have then that µν + c1 sin2(θ) = 0, which yields λ = ν.This is of course a contradiction. If λµ+ c1 = λν + c2 cos2(θ) = 0, theneν lies in sp{U} and hence eλ lies in D−1, which is a contradiction. Weconclude that if λµ + c1 = 0, then {eλ, eµ} ⊂ D1, ν = 0 and eν ∈ D−1.Analogously one can show that {eλ, eµ} ⊂ D−1, ν = 0 and eν ∈ D1, if


λµ+ c2 = 0. Suppose now that λµ = 0 = λν + c1 sin2(θ), then we havethat eλ ∈ D1, eν ∈ sp{U} and eµ ∈ D−1. Equation (4.43) yields eν ∈ D1,which is of course a contradiction because sp{U} and D1 are orthogonal.Analogously we obtain a contradiction if λµ = 0 = λν + c2 cos2(θ).Finally suppose that λν + c1 sin2(θ) = 0 = λµ+ c2 cos2(θ), then we havethat eλ ∈ sp{U}, eν ∈ D1 and eµ ∈ D−1. Equations (4.42) and (4.43)yield now that νµ = 0 and so we have that cos(θ) = 0 or sin(θ) = 0.This is a contradiction. From the previous we can conclude that U = 0,if S has at least different eigenvectors and hence we have a contradiction,because we have assumed that U 6= 0. So we obtain the proposition.

Theorem 4.5.5. LetM be a semi-parallel hypersurface ofM(c1)×M(c2).Then there are three possibilities:

1. c1 + c2 = 0 and M is a totally umbilical hypersurface,

2. M is an open part of M × M(c2), where M is semi-parallelhypersurface of M(c1),

3. M is an open part of M(c1) × M , where M is semi-parallelhypersurface of M(c2).

Proof. Let M be a semi-parallel hypersurface of M(c1) ×M(c2) withshape operator S. According to the previous proposition, there are threepossible forms of S to consider. In the first case, M is totally umbilicalby definition. This gives case (1) of the theorem. In the second casewe have that sin(θ) = 0, i.e. dimD1 = n1 − 1 and dimD−1 = n2 andhence we have that M is an open part of M ×M(c2), where M is ahypersurface of M(c1) with shape operator S. Since M(c1) is a totallygeodesic submanifold of M(c1)×M(c2), we have that the shape operatorS of M in M(c1) satisfies SX = SX for every X ∈ D1, such that S takesthe form of a shape operator of semi-parallel hypersurfaces in M(c1).The third case can be analogously proven as the second case.

As corollary we obtain now the parallel hypersurface of M(c1)×M(c2).

Theorem 4.5.6. Let M be a parallel hypersurface of M(c1) ×M(c2).Then there are two possibilities;

1. M is an open part of a Riemannian product M ×M(c2), where Mis a parallel hypersurface of M(c1).


2. M is an open part of a Riemannian product M(c1)×M , where Mis a parallel hypersurface of M(c2).

Chapter 5

PMC surfaces in productspaces

5.1 Introduction

We say that a submanifold of a Riemannian manifold is a parallel meancurvature (PMC) submanifold if and only if the mean curvature vectorfield H is parallel as the section of the normal bundle. Of course triviallywe have that every minimal submanifold of a Riemannian manifold is aPMC submanifold. One can also show that a minimal submanifold ofa hypersphere Sn in the Euclidean space En+1 has a parallel non-zeromean curvature vector field when it is considered as a submanifold ofthe Euclidean space En+1. Another example of a PMC submanifold is ahypersurface of a Riemannian manifold with constant mean curvature(CMC). In this chapter we will focus our attention on PMC surfacesof product space M(c1) ×M(c2), i.e. on Riemannian submanifolds ofdimension two with parallel mean curvature vector field.

But let us first consider the classical topic of CMC surfaces in threedimensional manifolds, which appear as critical points of a variationalproblem. Namely the CMC surfaces minimize the surface area withor without volume constraint. The unrestricted case coincides to zeromean curvature, i.e. the minimal surfaces. In the past especially CMCsurfaces in space forms were studied. In this case CMC surfaces withsome global assumptions like compactness, completeness, embeddedness,. . . were studied. One of the first assumptions on CMC surfaces was the

95

96 PMC SURFACES IN PRODUCT SPACES

assumption on genus of the CMC surface. In 1951 H. Hopf studied closedoriented CMC surfaces with genus 0, i.e. surfaces homeomorphic to thesphere, in the Euclidean space E3. He showed in [32], using the equationof Codazzi, the existence of a holomorphic differential on a CMC surfacein E3 and that this holomorphic differential vanishes if and only if thesurface is umbilical, i.e. the surface is a open part of a round sphere.Hence he obtained that a closed oriented CMC surface of genus 0 inE3 is an open part of a round sphere. This result was later extendedto CMC surfaces of 3-dimensional space forms by S-S. Chern in [12].The problem of classifying closed oriented CMC surfaces with genus 0 iscalled the Hopf problem. A holomorphic differential on a CMC surfacewill be called a Hopf differential.

In the last decade the study of CMC surfaces in three dimensionalmanifolds was extended to the study of CMC surfaces in simply connectedhomogeneous 3-dimensional ambient spaces. This class of manifoldsconsists of space forms (i.e. E3, S3 and H3), the five remaining Thurston3-dimensional geometries (i.e. the product spaces S2 × R and H2 × R,the Heisenberg group Nil3, the universal covering of PSL2(R) and theLie group Sol3), 3-dimensional Berger spheres and Lie groups with left-invariant metrics. This extension started after the work of Rosenbergand Meeks in [43],[42] and [47], where they established many results oncomplete minimal surfaces in M2 × R and after the work of Rosenbergand Abresch in [1] and [2], where they established results on theHopf problem in homogeneous 3-dimensional spaces with 4-dimensionalisometry group. Rosenberg and Abresch showed that there exists aholomorphic differential on CMC surfaces in 3-dimensional homogeneousspaces with 4-dimensional isometry group, which is a perturbationof the classical holomorphic differential of Hopf for CMC surfaces inspace forms, and also completely classified CMC surfaces for which thisholomorphic differential vanishes. We would like to refer to [27], where analternative proof is given for these results on the Hopf problem by usingthe integrability condition for surfaces in 3-dimensional homogeneousspaces(see [13]). In [27] one can also find a survey on the global theoryof CMC surfaces in homogeneous 3-dimensional spaces.

The natural generalization of CMC surfaces to higher co-dimensionalsurfaces is the notion of PMC surfaces. There exists also an extensivelystudy of PMC surfaces in space forms. In this respect we would like to

INTRODUCTION 97

mention the paper [31] of Hoffman, where he considered PMC surfacesin En. He showed for example in this paper that a closed orientedPMC surface with genus 0 is a pseudo-umbilical surfaces which liesin a hypersphere. The complete classification of PMC surfaces in theEuclidean space was given by Chen in [7] and , independently, by Yauin [61]. They showed that a PMC surface in the Euclidean space Enis a minimal surface of En, a minimal surface of a hypersphere of En,a CMC surface of E3 or a CMC surface lying in a hypersphere of anaffine 4-dimensional subspace of En. Similar result hold also for PMCsurfaces in spheres and the hyperbolic spaces ([8]). Also the study ofPMC surfaces in space forms was extended to the study of PMC surfacesin homogeneous spaces and in particular to the study of PMC surfaces ofa product space of two space forms. One of the first papers on this topicwas the paper of Torralbo and Urbano. They studied PMC surfaces inproduct spaces of two constant sectional curvature surfaces using thetwo natural Kähler structures of this ambient space in [56]. They areable to construct two Hopf differentials on these products using the twonatural Kähler structures that these products possess. Using these Hopfdifferentials they can classify PMC spheres in these products. We wouldlike to mention also the paper [4] in which the authors consider PMCsurfaces immersed in M(c) × R, where M(c) is a space form. In thispaper they generalize the results of Abresch and Rosenberg about CMCsphere to PMC spheres in M(c) × R. They use in this respect a Hopfdifferential which they defined in [3].

In this chapter we would like to give the construction of a Hopf differentialof a PMC surface of a product space M(c1)×M(c2) of two space forms.We mention that this Hopf differential will be constructed using theproduct structure F of the product space M(c1)×M(c2). We will alsoshow that this Hopf differential can be used in order to study PMCsurfaces with non-negative constant sectional curvature.

This chapter is organized as follows. In the first section we will give thedefinitions and notations that will be used in the next sections. In thesecond section we will give the Hopf differential for PMC surfaces inproduct spaces M(c1)×M(c2) and proof that this Hopf differential isindeed holomorphic. In the last section section we will show that thisHopf differential can be used in the study of PMC surfaces of productspaces with non-negative sectional curvature. We will namely give Simons’


type formula for PMC surfaces of M(c1)×M(c2), extending the workFetcu and Rosenberg.

5.2 Preliminaries and definitions

In this chapter we will consider surfaces isometrically immersed inM(c1) ×M(c2) with parallel mean curvature vector. So let M2 be asurface isometrically immersed inM(c1)×M(c2) with second fundamentalform σ and normal connection ∇⊥. Denote by ∇ and ∇ the Levi-Civitaconnection ofM(c1)×M(c2) andM2, respectively. Let F be the productstructure of M(c1)×M(c2). The mean curvature vector H is defined as

12(σ(e1, e1) + σ(e2, e2)),

where {e1, e2} is an orthonormal frame on M2. We say that a surfacehas parallel mean curvature vector if and only if ∇⊥XH = 0 for everyX ∈ TM2. We call surfaces with parallel mean curvature vector PMCsurfaces. Let us now consider isothermal coordinates (u, v) on M2

induced by the metric g of M2 and so the metric g of M2 can bewritten as λ2(du2 + dv2), where λ is a non-negative function. Associatedto these isothermal coordinates we consider the conformal parameterz = u + iv, we will consider the complex operators ∂ = 1√

2(∂u + i∂v)and ∂ = 1√

2(∂u − i∂v). Then we have that g(∂, ∂) = g(∂, ∂) = 0 andg(∂, ∂) = λ2 and that the Levi-Civita connection ∇ of M2 is given by∇∂∂ = 2∂(λ)

λ ∂ and ∇∂∂ = 0. For a symmetric (1, 1) tensor field S overM we have

g(S∂, ∂) = λ2

2 trS.

Moreover one can find that

S∂ = trS2 ∂ + g(S∂, ∂)

λ2 ∂. (5.1)

Analogously we have that

σ(∂, ∂) = λ2H.

In the next chapter we will need the Codazzi for surfaces ofM(c1)×M(c2)in order to show that our quadratic differential is holomorphic. We give

HOPF DIFFERENTIAL FOR PMC SURFACES IN M(C1)×M(C2) 99

therefore the equation of Codazzi in terms of ∂ and ∂:

(∇σ)(∂, ∂, ∂)− (∇σ)(∂, ∂, ∂) = ∇⊥∂σ(∂, ∂)− 2σ(∇∂∂, ∂)−∇⊥∂ (∂, ∂)

+ σ(∇∂∂, ∂) + σ(∂,∇∂∂)

= ∇⊥∂σ(∂, ∂)−∇⊥∂ (λ2H) + 2∂(λ)λ

H

= ∇⊥∂σ(∂, ∂)− λ2∇⊥∂H.

Hence we obtain the equation of Codazzi is given as follows in terms of∂ and ∂

∇⊥∂σ(∂, ∂) = λ2∇⊥∂H+h

(c1(∂ ∧ ∂)I + f

2 ∂ − c2(∂ ∧ ∂)I − f2 ∂

). (5.2)

In the next section we will also need the following equation that can bededuced by direct calculations:

(∇X(X1 ∧X2))Z = (∇XX1 ∧X2)Z + (X1 ∧∇XX2)Z, (5.3)

where ∇X(X1 ∧X2) is the covariant derivative of (X1 ∧X2).

5.3 Hopf differential for PMC surfaces inM(c1)×M(c2)

In this section we will introduce a Hopf differential for PMC surfaces inM(c1)×M(c2). In [3] and [4] Alencar, do Carmo and Tribuzy introduceda quadratic form Q on PMC surfaces of M(c)× R which is given by

Q(X,Y ) = 2g(AHX,Y )− cg(X, ∂t)g(Y, ∂t),

where ∂t is the vertical vector corresponding to the second factor Rof M(c) × R. They showed that the (2, 0)−part Q(2,0) = ψdz2 of thequadratic form Q of PMC surfaces is holomorphic. The means that thecomplex function

ψ = Q(2,0)(∂, ∂) = 2g(AH∂, ∂)− cg(∂, ∂t)2

is holomorphic, i.e. ∂(ψ) = 0. This is an extension of the theorem givenby Abresch and Rosenberg in [1]. Moreover this quadratic form is just


an extension of the quadratic form of Abresch and Rosenberg to highercodimensional surfaces of M(c)× R. They use this theorem to classifyall the PMC spheres of M(c)× R. Here we would like to generalize thetheorem about the holomorphic differential of PMC surfaces of M(c)×Rto the case of PMC surfaces ofM(c1)×M(c2). We introduce therefore thefollowing quadratic Qc1,c2 form on a PMC surface M2 of M(c1)×M(c2):

Qc1,c2(X,Y ) = 2|H|2g(σ(X,Y ), H)

+ c1g((P1X ∧ P1H)H,Y ) + c2g((P2X ∧ P2H)H,Y ),

where P1 = I+F2 and P2 = I−F

2 . We remark also that the length of H ofa PMC surface is constant. We show now that the Q(2,0)

c1,c2−part of thequadratic form Qc1,c2 is holomorphic, i.e. ∂Qc1,c2(∂, ∂) = 0:

∂Qc1,c2(∂, ∂) = 2|H|2g(∇⊥∂σ(∂, ∂), H) + 2c1

(g((P1(∇∂∂) ∧ P1H)H, ∂)

+g((P1∂ ∧ P1(∇∂H))H, ∂))

+ 2c2(g((P2(∇∂∂) ∧ P2H)H, ∂)

+ g((P2∂ ∧ P2(∇∂H))H, ∂))

= 2|H|2g(∇⊥∂σ(∂, ∂), H)

+ 2c1(λ2g((P1H ∧ P1H)H, ∂)− g((P1∂ ∧ P1(SH∂))H, ∂)

)+ 2c2

(λ2g((P2H ∧ P2H)H, ∂)− g((P2∂ ∧ P2(SH∂))H, ∂)

)= 2|H|2g(∇⊥

∂σ(∂, ∂), H)

− trSHc1g((P1∂ ∧ P1∂)H, ∂)− trSHc2g((P2∂ ∧ P2∂)H, ∂)

= 2|H|2(g(∇⊥∂σ(∂, ∂), H)

− c1g((P1∂ ∧ P1∂)H, ∂)− c2g((P2∂ ∧ P2∂)H, ∂))

= 0.

HOPF DIFFERENTIAL FOR PMC SURFACES IN M(C1)×M(C2) 101

In the previous calculation we have used the symmetries of the wedgeproduct ∧, the symmetry of the product structure F , equations (5.1)and (5.3) and the equation of Codazzi (5.2).

Theorem 5.3.1. Let M2 be a PMC surface of M(c1) ×M(c2). Thenthe (2, 0)−part Q(2,0)

c1,c2 of the quadratic form Qc1,c2 is holomorphic.

Remark 5.3.2. We mention that holomorphic differentials on PMCsurfaces of M(c)×M(−c) and M(c)× R have been found in the paperof Lira and Vitório in [15]. In the same paper they showed that PMCspheres in M3(c) × R lie in a totally geodesic cylinder M2(c) × R orin a leaf M3(c)× R. This theorem was generalized to PMC spheres inM(c)×R with dimension ofM(c) ≥ 2 in [4]. They showed first, under noglobal conditions, that PMC surfaces of M(c)× R are pseudo-umbilical,i.e. AH is an multiple of the identity, and lie in M(c) or that the PMCsurfaces lie in a totally geodesic submanifold of M(c)× R of dimension5. Using this fact they are able to classify PMC spheres in M(c)× R.Remark 5.3.3. Let us calculate the quadratic form Qc1,c2 in the case ofPMC surfaces of M(c)× R in terms of the vertical vector field ∂t.

Qc1,c2 = 2|H|2g(σ(X,Y ), H) + cg((P1X ∧ P1H)H,Y )

= 2|H|2g(σ(X,Y ), H)

+ cg(((X − g(X, ∂t)∂t) ∧ (H − g(H, ∂t)∂t)), Y )

= 2|H|2g(σ(X,Y ), H) + c(|H|2g(X,Y )− g(H, ∂t)2g(X,Y )

− |H|2g(X, ∂t)g(Y, ∂t))

= |H|2Q(X,Y ) + c(|H|2g(X,Y )− g(H, ∂t)2g(X,Y )).

So we obtain that our quadratic form Qc1,c2 is equal to |H|2Q+c(|H|2g−g(H, ∂t)2g) for surfaces in M(c) × R. But one can easily see that the(2, 0)−part Q(2,0)

c1,c2 of Qc1,c2 is equal to the (2, 0)−part Q(2,0) of Q modulothe factor |H|2, which is constant for PMC surfaces.


5.4 Non-negative PMC surfaces inM(c1)×M(c2)

In this section we would like to generalize the Simons’ type formula ofFetcu and Rosenberg for non-minimal PMC surfaces in M(c)× R to amore general Simons’ type formula for non-minimal PMC surfaces ofM(c1)×M(c2). Simons discovered in [49] a fundamental formula for theLaplacian of the second fundamental form for minimal submanifolds ofRiemannian manifolds. This formula permit him to proof certain resultson minimal submanifolds of a sphere and Euclidean space. This formulawas later generalized by Nomizu and Smyth for CMC hypersurfaces ofa space form in [45]. Using this formula they established many resultson complete CMC hypersurfaces with non-negative sectional curvature.Later Smyth generalized this formula to PMC surfaces of a space form(see [50]). In fact he obtained a Simons’ type formula for submanifoldsof a space form with an arbitrary parallel normal direction ξ for whichthe associated shape operator Sξ has constant trace. His proof for thisSimons’ type formula is based on the Codazzi equation for space forms.Namely, the Codazzi equation for submanifolds of space forms tells usthat the shape operator Sξ associated to ξ is a Codazzi tensor, i.e.

(∇XSξ)Y = (∇Y Sξ)X,

where (∇Sξ) is the covariant derivative of Sξ. In fact the condition onthe trace of the shape operator is superfluous. In [59] Wegner showedthe following theorem:

Theorem 5.4.1. Let M be a Riemannian manifold of dimension n, anddenote by g, | |, ∇, R, grad, div, ∆ and tr, the metric, the norm, theLevi-Civita connection on (M, g), the curvature tensor associated to ∇,the gradient, the divergence, the Laplacian and the trace operators on M ,respectively. Let A be a tensor field of type (1, 1) that is symmetric andsatisfies (∇XA)Y = (∇YA)X, i.e. A is a Codazzi tensor. Then

12∆(trA2) =

∑i<j

(λi−λj)2Kij+|∇A|2−|gradtr(A)|2+div(A(gradtr(A))),

(5.4)where λi are the eigenvalues of A corresponding to the eigenvectors ei andKij is the sectional curvature of M corresponding to the plane spannedby ei and ej.

NON-NEGATIVE PMC SURFACES IN M(C1)×M(C2) 103

We remark that equation (5.4) is a generalization of Smyth’s formulafor shape operators associated to parallel normal sections and constanttrace.

We would like to define now a symmetric (1, 1) tensor field on a PMCsurface M2 of M(c1) × M(c2) that is a Codazzi tensor. Like in theconstruction of the Hopf differential for PMC surfaces of M(c1)×M(c2),we can not use the shape operator associated to the mean curvature vectorH, because the shape operator SH of a PMC surface of M(c1)×M(c2)is not a Codazzi tensor. We use therefore a perturbation of the shapeoperator SH . Let us define a (1, 1) tensor field S on a PMC surface M2

of M(c1)×M(c2) by

g(SX, Y ) = Qc1,c2(X,Y )− trQc1,c2

2 g(X,Y ), (5.5)

where Qc1,c2 is the quadratic form defined in the previous section andhence we obtain from (5.5) that S is a symmetric and traceless (1, 1)tensor field over M2. In the next proposition we show that S is Codazzitensor field.

Proposition 5.4.2. The operator S on a PMC surfaces defined by (5.5)is a Codazzi tensor.

Proof. Let us denote by QS the quadratic form associated to S, i.e.QS(X,Y ) = g(SX, Y ). It is easy to see that the (2, 0)-part of thequadratic form QS is equal to the (2, 0)-part of the quadratic form Qc,c2

and hence we obtain, using Theorem 5.3.1, that the (2, 0)-part of QSis a holomorphic function and hence we have that QS is a holomorphicdifferential. From Lemma 6 of [44] we obtain now that S is a Codazzitensor, because QS is a holomorphic differential and traceless.

Using now equation (5.4) and the previous proposition, we obtain aSimons’ type formula for non-minimal PMC surfaces in M(c1)×M(c2).

Theorem 5.4.3. Let M2 be a non-minimal PMC surface of M(c1) ×M(c2) and S the symmetric and traceless (1, 1) tensor field defined by(5.5). Then

12∆tr(S2) = 2Ktr(S2) + |∇S|2,

where K is the Gaussian curvature of the PMC surface M2.


In the next proposition we would like to show that the (1, 1) tensor fieldS can be used in the study of complete PMC surfaces of M(c1)×M(c2)with non-negative Gaussian curvature. Let us first remark that thedeterminant of the shape operator SH associated to the mean curvaturevectorH can written in terms of the (1, 1) tensor field S, (1, 1)-tensor fieldf and the (1, 1) tensor field s. By a long and straightforward calculation,one can find

det(S H|H|

) = |H|2 − 14|H|2

(|S|2

2

− (c1 − c2)|H|2 + (c1 + c2)g(tH,H)4|H|2 tr(Sf) + c1 + c2

4|H|2 g(SsH, sH)+

(c1 − c2

4 + c1 + c24|H|2 g(tH,H)

)2(|f |2

2 − (trf)2

4

)

−(c2

1 − c22 + (c1 + c2)2g(tH,H)

16|H|2

)(g(fsH, sH)− trf

2 |sH|2)

+ (c1 + c2)2

16|H|2 |sH|4), (5.6)

where |S|2 = tr(S2). Hence we obtain that the Gauss curvature K ofa surface M2 of M(c1)×M(c2) can be written in terms of the secondfundamental form, the (1, 1) tensor field S, the (1, 1)-tensor field f , the(1, 1)-tensor field t and the (1, 1) tensor field s, because the Gaussiancurvature K is equal to

K = det(S H|H|

) +∑α

det(Sξα) + c1det(I + f

2 ) + c2det(I − f2 ), (5.7)

where {ξα} is an orthonormal frame orthogonal to the mean curvaturevector H. We are ready to proof the proposition.

Proposition 5.4.4. Let M2 be a complete non-minimal PMC surfaceof M(c1) × M(c2) with non-negative curvature. Then there are twopossibilities:

1. M2 is a flat surface;

NON-NEGATIVE PMC SURFACES IN M(C1)×M(C2) 105

2. the (2, 0)−part of the quadratic form Qc1,c2 on the PMC surfaceM2 of M(c1)×M(c2) vanishes.

Proof. Let us first show that the function |S| is bounded. We can writenow the Gaussian curvature in terms of the second fundamental form,the (1, 1)-tensor field S, the (1, 1) tensor field f , the (1, 1) tensor fieldt and the (1, 1) tensor field s using equations (5.6) and (5.7). Then,using the fact that M2 is a surface with non-negative Gaussian curvatureand the Chauchy-Schwarz inequality, one can find that the function |S|satisfies an inequality of the form

− 18|H|2 |S|

2 + a|S|+ b ≥ 0, (5.8)

where a and b are functions on M2 that can be bounded from above,because H is a constant and the (1, 1) tensor fields f, s and t are boundedtensor fields. Hence we obtain that |S| is bounded. So we have that|S|2 is a bounded function and 1

2∆|S|2 = 2K|S|2 + |∇S|2 ≥ 0, i.e.|S|2 is a subharmonic function that is bounded. We also have thatM2 is a parabolic space because M2 is complete and has non-negativeGaussian curvature(see [33]), i.e. M2 is a surface on which there existno nonconstant negative subharmonic function. Hence we obtain that|S| is a constant function and so we obtain, using Theorem 5.4.3, that2K|S|2 + |∇S|2 = 0. So we have that either K = 0 and |∇S| = 0 or that|S| = 0. By direct calculation one can show that the condition |S| = 0 isequivalent to the fact that the (2, 0)−part of the quadratic form Qc1,c2

vanishes.

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