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Holonomy theory of Finsler manifolds
Zoltan Muzsnay and Peter T. Nagy
Institute of Mathematics, University of DebrecenH-4010 Debrecen, Hungary, P.O.B. 12
E-mail: [email protected], [email protected]
Contents
1 Introduction 2
2 Preliminaries 42.1 Spray manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Horizontal distribution, covariant derivative and curvature . . . . . . . 42.1.2 Parallel translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Totally geodesic and auto-parallel submanifolds . . . . . . . . . . . . . 72.1.4 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Finsler metric, its associated spray and parallel translation . . . . . . 82.2.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Projectively flat Finsler manifold . . . . . . . . . . . . . . . . . . . . . 10
2.3 Finsler holonomy represented on the indicatrix bundle . . . . . . . . . . . . . 11
3 Diffeomorphism groups and their tangent algebras 123.1 Diffeomorphism group of compact manifolds . . . . . . . . . . . . . . . . . . . 16
4 Curvature algebra 164.1 Curvature vector fields at a point . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Constant curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Infinite dimensional curvature algebra . . . . . . . . . . . . . . . . . . . . . . 22
5 Holonomy algebra 245.1 Fibred holonomy group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Infinitesimal holonomy algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Holonomy algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Finsler surfaces with hol∗(x) = Rx . . . . . . . . . . . . . . . . . . . . . . . . 27
6 Infinite dimensional infinitesimal holonomy algebra 296.1 Projective Finsler surfaces of constant curvature . . . . . . . . . . . . . . . . 296.2 Projective Finsler manifolds of constant curvature . . . . . . . . . . . . . . . 34
6.2.1 Totally geodesic and auto-parallel submanifolds . . . . . . . . . . . . . 34
7 Dimension of the holonomy group 35
8 Maximal holonomy 388.1 Holonomy group as a subgroup of the diffeomorphism group of the indicatrix 388.2 The group Diff∞+ (S1) and the Fourier algebra . . . . . . . . . . . . . . . . . . 398.3 Holonomy of the standard Funk plane and the Bryant-Shen 2-spheres . . . . 40
2000 Mathematics Subject Classification: 53B40, 53C29Key words and phrases: Finsler geometry, holonomy.This research was partially supported by the EU FET FP7 BIOMICS project, contract number CNECT-
318202.
1
1 Introduction
The notion of the holonomy group of a Riemannian or Finslerian manifold can be intro-duced in a very natural way: it is the group generated by parallel translations along loops.In contrast to the Finslerian case, the Riemannian holonomy groups have been extensivelystudied. One of the earliest fundamental results is the theorem of Borel and Lichnerowicz [3]from 1952, claiming that the holonomy group of a simply connected Riemannian manifoldis a closed Lie subgroup of the orthogonal group O(n). By now, the complete classificationof Riemannian holonomy groups is known. The holonomy group of a Finsler manifold is thesubgroup of the diffeomorphism group of an indicatrix, generated by canonical homogeneous(nonlinear) parallel translations along closed loops. Before our investigation (c.f. [24], [25],[26], [27], [28], [29]), the holonomy groups of non-Riemannian Finsler manifolds have beendescribed only in special cases: for Berwald manifolds there exist Riemannian metrics withthe same holonomy group (cf. Z. I. Szabo, [39]), for positive definite Landsberg manifoldsthe holonomy groups are compact Lie groups consisting of isometries of the indicatrix withrespect to an induced Riemannian metric (cf. L. Kozma, [19], [20]). A thorough study ofholonomy groups of homogeneous (nonlinear) connections was initiated by W. Barthel in hisbasic work [2] in 1963; he gave a construction for a holonomy algebra of vector fields on thetangent space. A general setting for the study of infinite dimensional holonomy groups andholonomy algebras of nonlinear connections was initiated by P. Michor in [23]. However theintroduced holonomy algebras could not be used to estimate the dimension of the holonomygroup since their tangential properties to the holonomy group were not clarified.In this paper we construct and investigate tangent Lie algebras to the holonomy group andto show that the dimension of these tangent algebras in many cases is greater then thepossible dimensions of Riemannian holonomy groups.In the second section we collect the necessary definitions and constructions of spray andFinsler geometry. The third section is devoted to the investigation of tangential propertiesof subalgebras of the Lie algebra of vector fields on a manifold to the infinite dimensionaldiffeomorphis group of this manifold. Particularly we consider the case if the manifold iscompact, in this case the diffeomorphism group is an infinite dimensional Lie group modeledon the Lie algebra of vector fields on the manifold.In Section 4 we introduce the notion of curvature algebra of a Finsler manifold consistingof of tangent vector fields on the indicatrix, which is a generalization of the matrix groupgenerated by curvature operators of a Riemannian manifold. We show that the vector fieldsbelonging to the curvature algebra are tangent to one-parameter families of diffeomorphismscontained in the holonomy group. We prove that for a positive definite non-RiemannianFinsler manifold of non-zero constant curvature with dimension n > 2 the dimension of thecurvature algebra is strictly greater than the dimension of the orthogonal group acting onthe tangent space and hence it can not be a compact Lie group. In addition, we provide anexample of a left invariant singular (non y-global) Finsler metric of Berwald-Moor-type onthe 3-dimensional Heisenberg group which has infinite dimensional curvature algebra andhence its holonomy is not a (finite dimensional) Lie group. These results give a positive an-swer to the following problem formulated by S. S. Chern and Z. Shen in [8] (p. 85): Is therea Finsler manifold whose holonomy group is not the holonomy group of any Riemannianmanifold?Section 4 contains construction of further tangent Lie algebras to the holonomy group con-sisting of of tangent vector fields on the indicatrix, namely the infinitesimal holonomy algebraand the holonomy algebra of a Finsler manifold. Our goal is to make an attempt to findthe right notion of the holonomy algebra of Finsler spaces. The holonomy algebra shouldbe the largest Lie algebra such that all its elements are tangent to the holonomy group. Inour attempt we are building successively Lie algebras having the tangent properties. Wedefine the infinitesimal holonomy algebra by the smallest Lie algebra of vector fields on anindicatrix, containing the curvature vector fields and their horizontal covariant derivativeswith respect to the Berwald connection and prove the tangential property of this Lie algebra
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to the holonomy group. At the end we introduce the notion of the holonomy algebra of aFinsler manifold by all conjugates of infinitesimal holonomy algebras by parallel translationswith respect to the Berwald connection. We prove that this holonomy algebra is tangent tothe holonomy group. The question of whether the holonomy algebra introduced in this wayis the largest Lie algebra, which is tangent to the holonomy group, is still open.In Section 6 we construct for interesting classes of locally projectively flat Finsler surfacesand manifolds of non-zero constant curvature infinite dimensional subalgebras in the tan-gent infinitesimal holonomy algebras. From the viewpoint of non-Euclidean geometry themost important Riemann-Finsler manifolds are the projectively flat spaces of constant flagcurvature. We will turn our attention to non-Riemannian projectively flat Finsler manifoldsof non-zero constant flag curvature. We consider the following classes of locally projectivelyflat non-Riemannian Finsler manifolds of non-zero constant flag curvature:
1. Randers manifolds,
2. manifolds having a 2-dimensional subspace in the tangent space at some point, onwhich the Finsler norm is an Euclidean norm,
3. manifolds having a 2-dimensional subspace in the tangent space at some point, onwhich the Finsler norm and the projective factor are linearly dependent.
The first class consists of positively complete Finsler manifolds of negative curvature, thesecond class contains a large family of (not necessarily complete) Finsler manifolds of nega-tive curvature, and the third class contains a large family of not necessarily complete Finslermanifolds of positive curvature. The metrics belonging to these classes can be considered as(local) generalizations of a one-parameter family of complete Finsler manifolds of positivecurvature defined on S2 by R. Bryant in [Br1], [Br2] and on Sn by Z. Shen in [37], Example7.1. We prove that the holonomy group of Finsler manifolds belonging to these classes andsatisfying some additional technical assumption is infinite dimensional. In Section 7 we areinvestigating the holonomy group of an arbitrary locally projectively flat Finsler manifoldsof constant curvature. Our aim is to characterize all locally projectively flat Finsler man-ifolds with finite dimensional holonomy group. To obtain such a characterization, we willinvestigate the dimension of the infinitesimal holonomy algebra. We prove that if (M,F) isa non-Riemannian locally projectively flat Finsler manifolds of nonzero constant curvature,then its infinitesimal holonomy algebra is infinite dimensional. Using this general result andthe tangent property of the infinitesimal holonomy algebra we obtain the characterization:The holonomy group of a locally projectively flat Finsler manifold of constant curvature isfinite dimensional if and only if it is a Riemannian manifold or a flat Finsler manifold.Section 8 is devoted to show that the topological closure of the holonomy group of a certainclass of simply connected, projectively flat Finsler 2-manifolds of constant curvature is not afinite dimensional Lie group, and we prove that its topological closure is the connected com-ponent of the full diffeomorphism group of the circle. Until now, perhaps because of technicaldifficulties, not a single infinite dimensional Finsler holonomy group has been described. Inthis paper we provide the first such a description. This class of Finsler 2-manifolds containsthe positively complete standard Funk plane of constant negative curvature (positively com-plete standard Funk plane), and the complete irreversible Bryant-Shen-spheres of constantpositive curvature ([37], [5]). We obtain that for every simply connected Finsler 2-manifoldthe topological closure of the holonomy group is a subgroup of Diff∞+ (S1). That means thatin the examples mentioned above, the closed holonomy group is maximal. In the proof weuse our constructive method developed in Section 6 for the study of Lie algebras of vectorfields on the indicatrix, which are tangent to the holonomy group.
3
2 Preliminaries
2.1 Spray manifolds
A spray on a manifold M is a smooth vector field S on TM := TM \{0} expressed in astandard coordinate system (xi, yi) on TM as
S = yi∂
∂xi− 2Gi(x, y)
∂
∂yi, (1)
where the functions Gi(x, y) of local coordinates (xi, yi) on TM satisfy
Gi(x, λy) = λ2Gi(x, y), λ > 0. (2)
A manifold M with a spray S is called a spray manifold (M,S), cf. [34], Chapter 4.A curve c(t) is called geodesic of the spray manifold (M,S) if its coordinate functions ci(t)satisfy the system of 2nd order ordinary differential equations
ci(t) + 2Gi(c(t), c(t)) = 0, (3)
where the functions Gi(x, y) are called the geodesic coefficients the spray manifold (M,S).
2.1.1 Horizontal distribution, covariant derivative and curvature
Let (TM, π,M) and (TTM, τ, TM) be the first and the second tangent bundle of the man-ifold M , respectively, and let VTM⊂TTM be the (integrable) vertical distribution on TMgiven by VyTM := Kerπ∗,y. The horizontal distribution HTM ⊂ TTM associated to thespray manifold (M,S) is the image of the horizontal lift which is a vector space isomorphismly : TxM → HyTM for every x ∈M and y ∈ TxM defined by
ly
( ∂
∂xi
)=
∂
∂xi−Gki (x, y)
∂
∂yk, where Gij =
∂Gi
∂yj(4)
in the coordinate system (xi, yi) of TM . The horizontal distribution is complementary tothe vertical distribution, hence we have the decomposition TyTM = HyTM ⊕ VyTM . Theprojectors corresponding to this decomposition will be denoted by h : TTM → HTM andv : TTM → VTM .The vertical distribution over the slit tangent bundle TM = TM \{0} will be denoted by(VTM, τ, TM) and the pull-back bundle of (TM, π,M) corresponding to the map π : TM →M by (π∗TM, π, TM). Clearly, the mapping
(x, y, ξi∂
∂yi) 7→ (x, y, ξi
∂
∂xi) : VTM → π∗TM (5)
is a canonical bundle isomorphism. In the following we will use the isomorphism (5) for theidentification of these bundles.Let X∞(M) be the vector space of smooth vector fields on the manifold M and X∞(TM)the vector space of smooth sections of the bundle (VTM, τ, TM). The horizontal covariant
derivative of a section ξ ∈ X∞(TM) by a vector field X ∈ X∞(M) is given by
∇Xξ := [h(X), ξ].
The horizontal covariant derivative of ξ(x, y) = ξi(x, y) ∂∂yi by X(x) = Xi(x) ∂
∂xi can beexpressed as
∇Xξ =
(∂ξi(x, y)
∂xj−Gkj (x, y)
∂ξi(x, y)
∂yk+Gijk(x, y)ξk(x, y)
)Xj ∂
∂yi, (6)
4
where Gijk :=∂Gi
j
∂yk.
Defining the horizontal covariant derivative
∇Xφ =
(∂φ
∂xj−Gkj (x, y)
∂φ(x, y)
∂yk
)Xj
of a smooth function φ : TM → R, the horizontal covariant derivation (6) can be extended tosections of the tensor bundle over (π∗TM, π, TM), using the canonical bundle isomorphism(5).The curvature tensor field
K(x,y)(X,Y ) := v[Xh, Y h
], X, Y ∈ TxM. (7)
on the pull-back bundle (π∗TM, π, TM) of the spray manifold (M,S) in a local coordinatesystem (xi, yi) of TM is given by
K(x,y) = Kijk(x, y)dxj ⊗ dxk ⊗ ∂
∂xi,
where
Kijk(x, y) =
∂Gij(x, y)
∂xk− ∂Gik(x, y)
∂xj+Gmj (x, y)Gikm(x, y)−Gmk (x, y)Gijm(x, y). (8)
The curvature tensor field characterizes the integrability of the horizontal distribution.Namely, if the horizontal distribution HTM is integrable, then the curvature is identicallyzero.
2.1.2 Parallel translation
For a spray manifold (M,S) the parallel vector fields X(t) = Xi(t) ∂∂xi along a curvec(t) are
defined by the solutions of the differential equation
DcX(t) :=(dXi(t)
dt+Gij(c(t), X(t))cj(t)
) ∂
∂xi= 0. (9)
Using the relations (2) and Euler theorem on homogeneous functions we see that the func-tions Gij(x, y) are positive homogeneous of first order with respect to the variable y, andhence Dc(λX(t)) = λDcX(t) for any λ ≥ 0. The differential equation (9) can be expressedby the horizontal covariant derivative (6) as follows: a vector field X(t) = Xi(t) ∂
∂xi along acurve c(t) is parallel if it satisfies the equation
∇cX(t) =(dXi(t)
dt+Gij(c(t), X(t))cj(t)
) ∂
∂xi= 0. (10)
Clearly, for any X0 ∈ Tc(0)M there is a unique parallel vector field X(t) along the curve csuch that X0 =X(0). Moreover, if X(t) is a parallel vector field along c, then λX(t) is alsoparallel along c for any λ ≥ 0. Then the homogeneous (nonlinear) parallel translation
τc : Tc(0)M → Tc(1)M along a curve c(t)
of the spray manifold (M,S) is defined by the positive homogeneous map τc : X0 7→ X1 givenby the value X1 = X(1) at t = 1 of the parallel vector field with initial value X(0) = X0.Since the parallel translation of a spray manifold (M,S) is determined by its horizontaldistribution HTM ⊂ TTM , a spray manifold can be considered as a particular case ofa fibered manifold equipped with an Ehresmann connection, (cf. [9]). An Ehresmannconnection of a fibered manifold is given by a horizontal distribution, which is complement
5
X1
X0
c(t)
Figure 1: Parallel translation
to the vertical distribution consisting of the tangent spaces of the fibers. For a spray manifoldthe fibered manifold is the tangent bundle of M and the horizontal distribution determinedby the horizontal lift ly : TxM → HyTM expressed by equation (4).The parallel translation can be introduced a very nice geometrical way with the help ofthe notion of horizontal distribution. Namely, we call a curve in TM horizontal if thetangent vectors of this curve are contained in the horizontal distribution HTM ⊂ TTM .Let now c(t) be a curve in the manifold M joining the points p and q. The horizontallift ch(t) = (c(t), Xi(t) ∂
∂xi ) of c(t) is the curve ch(t) in TM defined by the properties thatch(t) projects on c(t) and ch(t) is horizontal that is ch(t) ∈ Hc(t). This means according toequation (4) that
ci(t)∂
∂xi+d
dtXi(t)
∂
∂yi=
(∂
∂xi−Gki (x, y)
∂
∂yk
)ci(t),
i.e. the tangent vector of the lifted curve ch(t) is the horizontal lift of the tangent vectorci(t) ∂
∂xi of c(t). It follows that a vector field X(t) along a curve c(t) is parallel if and onlyif it is a solution of the differential equation
d
dt
(c(t), Xi(t)
∂
∂xi
)= lX(t)(c(t)), (11)
or equivalently X(t) satisfies the differential equation (9). Hence the parallel translationalong a curve c(t) joining the points p and q is the map τc : TpM → TqM determined by theintersection points of the horizontal lifts of the curve c(t) with the tangent spaces Tp andTq. The construction can be illustrated by the following figure:
v c w
M
TM
π
v w
ch
Figure 2: Geometric construction of the parallel translation
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2.1.3 Totally geodesic and auto-parallel submanifolds
A submanifold M in a spray manifold (M,S) is called totally geodesic if any geodesic of(M,S) which is tangent to M at some point is contained in M .
A totally geodesic submanifold M of (M,S) is called auto-parallel if the homogeneous (non-linear) parallel translations τc : Tc(0)M → Tc(1)M along curves in the submanifold M leave
invariant the tangent bundle TM and for every ξ ∈ X∞(TM) the horizontal Berwald co-
variant derivative ∇Xξ belongs to X∞(TM).
Let X,Y ∈ TxM be tangent vectors at x ∈ M and let K denote the curvature tensorof (M,S), (cf. equation (8)). The mapping y → K(X,Y )(x, y) : TxM 7→ TxM is calledcurvature vector field at x of the spray manifold (M,S).
Lemma 1 Let M be a totally geodesic submanifold in a spray manifold (M,S). The fol-lowing assertions hold:
(a) the spray S induces a spray S on the submanifold M ,
(b) M is an auto-parallel submanifold,
Proof. Assume that the manifolds M and M are k, respectively n = k + p dimensional.Let (x1, . . . , xk, xk+1, . . . , xn) be an adapted coordinate system, i. e. the submanifold M islocally given by the equations xk+1 = · · · = xn = 0. We denote the indices running on thevalues {1, . . . , k} or {k + 1, . . . , n} by α, β, γ or σ, τ , respectively. The differential equation(3) of geodesics yields that the geodesic coefficients Gσ(x, y) satisfy
Gσ(x1, . . . , xk, 0, . . . , 0; y1, . . . , yk, 0, . . . , 0) = 0
identically, hence their derivatives with respect to y1, . . . , yk are also vanishing. It followsthat Gσα = 0 and Gσα β = 0 at any (x1, . . . , xk, 0, . . . , 0; y1, . . . , yk, 0, . . . , 0). Hence the
induced spray S on M is defined by the geodesic coefficients
Gβ(x1, . . . , xk; y1, . . . , yk) = Gβ(x1, . . . , xk, 0, . . . , 0; y1, . . . , yk, 0, . . . , 0). (12)
The homogeneous (nonlinear) parallel translation τc : Tc(0)M → Tc(1)M along curves in thesubmanifold M and the horizontal covariant derivative on M with respect to the spray Scoincide with the translation and the horizontal covariant derivative on M with respect tothe spray S. Hence the assertions are true.
2.1.4 Holonomy
The notion of the holonomy group of an Ehresmann connection, or particularly of a spraymanifold can be introduced in a very natural way: it is the group generated by paralleltranslations along loops with respect to the associated connection, cf. [41], [18], pp. 82–86.The holonomy properties of a spray manifold depends essentially on its curvature properties.This can be easily understood by considering the geometric construction of the paralleltranslation.
Let (M,S) be a spray manifold, and let us assume that M is connected. We choose afixed base point p ∈ M . For each closed piecewise smooth curve c : [0, 1] → M through pthe parallel translation τc : TpM → TpM along the curve c : [0, 1]→M is a diffeomorphismof the tangent space TpM . All these diffeomorphisms form together the holonomy group atthe point p, a subgroup of the diffeomorphism group of TpM . Clearly, the holonomy groupdepends on the base point p only up to conjugation, and therefore the holonomy groups atdifferent points of M are isomorphic.• Case K ≡ 0. Let us consider a point p ∈ M , a tangent vector v ∈ TpM at p, and an
7
p
v
TM
M
v
Figure 3: Trivial holonomy: R ≡ 0
arbitrary closed curve c on Mstarting from p. If the curvatureidentically zero then the horizontaldistribution is integrable. There-fore TTM has a horizontal folia-tion, that is for every v ∈ TpMthere is a unique n-dimensionalsubmanifold Hv in TM such thatHv ∩ TpM = {v} and the tangentspaces are horizontal. On the Fig-ure 3 the level surface correspond-ing to v ∈ TpM is represented bydashed lines in tangent manifoldTM .The horizontal lift ch = l ◦ c of the
curve c starts at v and (because the integrability condition) stays on the horizontal sub-manifold Hv. Its endpoint is an element of Hv and also an element of TpM that is v.Consequently we can obtain that ch(0) = ch(1) = v, and the holonomy is trivial.• Case K 6≡ 0. Although the figure describing this case looks similar
p
v 6=τ(v)
TM
M
v
τ(v)
Figure 4: Nontrivial holonomy: R 6≡ 0
to that of Figure 3, the situationis quite different. The horizon-tal distribution is non-integrableand therefore there is no horizon-tal foliation. Let us consider apoint p ∈ M and a vector v ∈TpM . The smallest integrable dis-tribution Nv at v containing thehorizontal distribution is at least(n+ 1)-dimensional. The reach-able sets in TpM is at least 1-dimensional. In particular {v} (Nv ∩ TpM and there are otherelements TpM reachable from v.That is there are elements w ∈TpM and a horizontal curve ch ∈Nv such that ch(0) = v, ch(1) = w. Considering the projection of ch we can obtain a curvec := π ◦ ch such that c(0) = c(1) = p and τc(v) 6= v. Consequently we obtained that theholonomy is nontrivial. (Cf. [34], Remark 8.1.3.)
2.2 Finsler manifolds
2.2.1 Finsler metric, its associated spray and parallel translation
A Minkowski functional on a vector space V is a continuous function F , positively homo-geneous of degree two, i.e. F(λy) = λ2F(y) if λ > 0, smooth on V := V \ {0}, and for anyy ∈ V the symmetric bilinear form gy : V × V → R defined by
gy : (u, v) 7→ gij(y)uivj =1
2
∂2F(y + su+ tv)
∂s ∂t
∣∣∣t=s=0
is non-degenerate. If gy is positive definite for any y ∈ V then F is said positive definite and(V,F) is called positive definite Minkowski space. A Minkowski functional F is called semi-Euclidean if there exists a symmetric bilinear form 〈 , 〉 on V such that gy(u, v) = 〈u, v〉 for
any y ∈ V and u, v ∈ V . A semi-Euclidean positive definite Minkowski functional is called
8
Euclidean.
A Finsler manifold is a pair (M,F) of an n-manifold M and a function F : TM → R(called Finsler metric, cf. [34]) defined on the tangent bundle of M , which is smooth onTM := TM \{0} and its restriction Fx = F|
TxMis a Minkowski functional on TxM for all
x ∈M . If the Minkowski functional Fx is positive definite on TxM for all x ∈M then (M,F)is called positive definite Finsler manifold. A point x ∈ M is called (semi-)Riemannian ifthe Minkowski functional Fx is (semi-)Euclidean.
We remark that in many applications the metric F is defined and smooth only on an opencone CM⊂TM \{0}, where CM=∪x∈MCxM is a fiber bundle over M such that each CxMis an open cone in TxM\{0}. In such case (M,F) is called singular (or non y-global) Finslerspace (cf. [34]).
The symmetric bilinear form
gx,y : (u, v) 7→ gij(x, y)uivj =1
2
∂2F2x(y + su+ tv)
∂s ∂t
∣∣∣t=s=0
, u, v ∈ TxM
is called the metric tensor of the Finsler manifold (M,F). The Finsler function is calledabsolutely homogeneous at x ∈ M , if Fx(λy) = |λ|Fx(y) for all λ ∈ R. If F is absolutelyhomogeneous at every x ∈M , then the Finsler manifold (M,F) is reversible.
The simplest non-Riemannian Finsler metrics are the Randers metrics firstly studied by G.Randers in [33]. A Finsler manifold (M,F) is called Randers manifold if the Finsler metricF can be expressed in the form F (x, y) = ax(y) + bx(y), where ax(y) =
√aij(x)yiyj is a
Riemannian metric and bx(y) = bi(x)yi is a nowhere zero 1-form.
The canonical spray of a Finsler manifold (M,F) is locally given by S = yi ∂∂xi−2Gi(x, y) ∂
∂yi ,where
Gi(x, y) :=1
4gil(x, y)
(2∂gjl∂xk
(x, y)− ∂gjk∂xl
(x, y))yjyk. (13)
are the geodesic coefficients. The geodesics of a Finsler manifold (M,F) are the geodesics ofthe canonical spray of (M,F) determined by the differential equations (3). The horizontalcovariant derivative with respect to the spray associated to the Finsler manifold (M,F) iscalled the horizontal Berwald covariant derivative, cf. equation (6). The corresponding ho-mogeneous (nonlinear) parallel translation τc : Tc(0)M → Tc(1)M along a curve c(t), given byequations (9) and (10), is called the canonical homogeneous (nonlinear) parallel translationof the Finsler manifold (M,F). Since the geodesic coefficients Gi(x, y) are differentiablefunctions on the slit tangent bundle TM = TM \ {0} the canonical homogeneous (nonlin-ear) parallel translation τc : Tc(0)M → Tc(1)M induces differentiable maps between the slittangent spaces Tc(0)M \ {0} and Tc(1)M \ {0}.
2.2.2 Curvature
The Riemannian curvature tensor field of the Finsler manifold (M,F) is the curvature tensorfield v
[Xh, Y h
]of the canonical spray manifold of (M,F) is defined on the pull-back bundle
(π∗TM, π, TM), (cf. equation (7)). If HTM is integrable, then the Riemannian curvatureis identically zero. According to equation (8) the expression of the Riemannian curvaturetensor R(x,y) = Rijk(x, y)dxj ⊗ dxk ⊗ ∂
∂xi is
Rijk(x, y) =∂Gij(x, y)
∂xk− ∂Gik(x, y)
∂xj+Gmj (x, y)Gikm(x, y)−Gmk (x, y)Gijm(x, y)
in a a local coordinate system. The manifold (M,F) has constant flag curvature λ ∈ R, iffor any x ∈M the local expression of the Riemannian curvature is
Rijk(x, y) = λ(δikgjm(x, y)ym − δijgkm(x, y)ym
). (14)
9
In this case the flag curvature of the Finsler manifold (cf. [8], Section 2.1 pp. 43-46) doesnot depend either on the point or on the 2-flag.
The Berwald curvature tensor field B(x,y) = Bijkl(x, y)dxj ⊗ dxk ⊗ dxl ⊗ ∂∂xi is
Bijkl(x, y) =∂Gijk(x, y)
∂yl=∂3Gi(x, y)
∂yj∂yk∂yl. (15)
The mean Berwald curvature tensor field E(x,y) = Ejk(x, y)dxj ⊗ dxk is the trace
Ejk(x, y) = Bljkl(x, y) =∂3Gl(x, y)
∂yj∂yk∂yl. (16)
The Landsberg curvature tensor field L(x,y) = Lijkl(x, y)dxj ⊗ dxk ⊗ dxl ⊗ ∂∂xi is
L(x,y)(u, v, w) = g(x,y)
(∇wB(x,y)(u, v, w), y
), u, v, w ∈ TxM.
According to Lemma 6.2.2, equation (6.30), p. 85 in [34], one has for u, v, w ∈ TxM
∇wg(x,y)(u, v) = −2L(x,y)(u, v, w).
Lemma 2 The horizontal Berwald covariant derivative of the tensor field
Q(x,y) =(δijgkm(x, y)ym − δikgjm(x, y)ym
)dxj ⊗ dxk ⊗ dxl ⊗ ∂
∂xi
vanishes.
Proof. For any vector field W ∈ X∞(M) we have ∇W y = 0 and ∇W IdTM = 0. Moreover,since L(x,y)(y, v, w) = 0 (cf. equation 6.28, p. 85 in [34]) we get the assertion.
2.2.3 Projectively flat Finsler manifold
A Finsler manifold (D,F) on an open subset D ⊂ Rn is said to be projectively flat, if allgeodesics of (D,F) are contained in straight lines of the affine space associated to Rn. AFinsler manifold (M,F) is said to be locally projectively flat, if for any point in p ∈M thereexists a local coordinate map x : U → Rn of a neighbourhood U ⊂ M of p such that theFinsler manifold induced by the Finsler function F on the image x(U) = D is projectivelyflat. The space Rn containing D is called to be projectively related to (M,F).
Let (M,F) be a locally projectively flat Finsler manifold and (x1, . . . , xn) : U → D a localcoordinate map corresponding to canonical coordinates of the space Rn which is projectivelyrelated to (M,F). Then the geodesic coefficients (13) are of the form
Gi(x, y) = P(x, y)yi, Gik =∂P∂yk
yi + Pδik, Gikl =∂2P∂yk∂yl
yi +∂P∂yk
δil +∂P∂yl
δik, (17)
where P is a 1-homogeneous function in y, called the projective factor of (M,F), (cf. [8],p. 63). Clearly, the intersections of 2-planes of An with the image D of the coordinate map(x1, . . . , xn) : U → D are images of totally geodesic submanifolds of (M,F).
Remark 3 The canonical homogeneous parallel translation τc : Tc(0)M → Tc(1)M in alocally projectively flat Finsler manifold (M,F) along curves c(t) contained in the domain ofthe coordinate system (x1, . . . , xn) are linear maps if and only if the projective factor P(x, y)is a linear function in y. Hence the non-linearity in y of the projective factor implies thatthe locally projectively flat Finsler manifold is non-Riemannian.
10
Projectively flat Randers manifolds with constant flag curvature were classified by Z.Shen in [36]. He proved that any projectively flat Randers manifold (M,F) with non-zero constant flag curvature has negative curvature. These metrics can be normalized by aconstant factor so that the curvature is − 1
4 . In this case (M,F) is isometric to the Finslermanifold defined by the metric function
F(x, y) =
√|y|2 − (|x|2|y|2 − 〈x, y〉2)
1− |x|2±(〈x, y〉
1− |x|2+
〈a, y〉1 + 〈a, x〉
)(18)
on the unit ball Dn ⊂ Rn, where a ∈ Rn is any constant vector with |a| < 1. According toLemma 8.2.1 in [8], p.155, the projective factor P(x, y) can be computed by the formula
P(x, y) =1
2F∂F∂xi
yi.
An easy calculation yields
± ∂F∂xi
yi =
(√|y|2 − (|x|2|y|2 − 〈x, y〉2)± 〈x, y〉
1− |x|2
)2
−(〈a, y〉
1 + 〈a, x〉
)2,
hence
P(x, y) =1
2
(±√|y|2 − (|x|2|y|2 − 〈x, y〉2) + 〈x, y〉
1− |x|2− 〈a, y〉
1 + 〈a, x〉
). (19)
2.3 Finsler holonomy represented on the indicatrix bundle
The notion of the holonomy group of a Riemannian or Finslerian manifolds is an adaptationof the corresponding notion of spray manifolds: it is the group generated by parallel transla-tions along loops with respect to the canonical associated connection. However, the paralleltranslation leaves invariant the indicatrix bundle, the holonomy group can be identified byits action on the indicatrix at the initial point. Hence the holonomy group can be consideredas a subgroup of the diffeomorphism group of the indicatrix.
X0=X1
(a) Trivial holonomy: the plane
X0
X1
(b) Nontrivial holonomy: sphere
Figure 5: Examples
In the Riemannian case, the holonomy groups have been extensively studied. One of theearliest fundamental results is the theorem of Borel and Lichnerowicz [3] from 1952, claimingthat the holonomy group of a simply connected Riemannian manifold is a closed Lie subgroupof the orthogonal group O(n). By now, the complete classification of Riemannian holonomygroups is known.
The holonomy properties of Finsler spaces is, however, essentially different from theRiemannian one, and it is far from being well understood. Compared to the Riemanniancase, only few results are known. The main difficulty comes from the fact that in thegeneral case the canonical connection of a Finsler manifold is neither linear nor metrical(that is the parallel translation is not necessarily preserving the metric). Only much weakerproperties are fulfilled: instead of the linearity it is only 1−homogeneous, and instead of
11
the metrical property it is preserving only the norm function. Nonetheless these propertiesallow us to consider the parallel translations as maps between the indicatrices and thereforethe holonomy group as a subgroup of the diffeomorphism group of the indicatrix.
Let (M,F) be an n-dimensional Finsler manifold. The indicatrix IxM at x ∈ M is ahypersurface of TxM defined by
IxM := {y ∈ TxM ; F(y) = ±1}.
If the Finsler manifold (M,F) is positive definite then the indicatrix IxM is a compacthypersurface in the tangent space TxM , diffeomorphic to the standard (n−1)-dimensionalsphere.In the sequel (IM,π,M) will denote the indicatrix bundle of (M,F) and i : IM ↪→ TM thenatural embedding of the indicatrix bundle into the tangent bundle (TM, π,M).
The parallel translation τc : Tc(0)M → Tc(1)M along a curve c : [0, 1] → R on a Finslermanifold (M,F) is defined by the parallel translation of the associated spray manifold (cf.Section 2.1.2). It is determined by vector fields X(t) along c(t) which are solutions of thedifferential equation (10). Since τc : Tc(0)M → Tc(1)M is a differentiable map between the
slit tangent spaces Tc(0)M and Tc(1)M preserves the value of the Finsler function, it inducesa map
τIc : Ic(0)M −→ Ic(1)M (20)
between the indicatrices. Since the parallel translation is 1-homogeneous, the parallel trans-lation τc is entirely characterized by the map τIc : we have τC(0) = 0 and for every non-zerovector v ∈ Tc(0)M we have
τc(v) = |v| · τIc(v
|v|
).
It follows from these observations that the holonomy group Hol(x) of the spray manifoldassociated to a Finsler manifold (M,F) (cf. Section 2.1.4) is uniquely determined by itsaction on the indicatrix in the tangent space TxM at the point x. Hence we can formulate
Definition 4 The holonomy group Hol(x) of a Finsler space (M,F) at x ∈ M is thesubgroup of the group of diffeomorphisms Diff(IxM) of the indicatrix IxM determined byparallel translation of IxM along piece-wise differentiable closed curves initiated at the pointx ∈M .
v
τ(v)
ϕ
Figure 6: Holonomy transformation induced on the indicatrix
We note that the holonomy group Hol(x) is a topological subgroup of the regular infinitedimensional Lie group Diff(IxM), but its differentiable structure is not known in general.(Diff(IxM) denotes the group of all C∞-diffeomorphism of IxM with the C∞-topology.)
3 Diffeomorphism groups and their tangent algebras
The group Diff(M) of all smooth diffeomorphisms of a differentiable manifold M is a regularinfinite dimensional Lie group modeled on the vector space Xc(M) of smooth vector fields
12
on M with compact support. The Lie algebra of the infinite dimensional Lie group Diff(M)is the vector space Xc(M), equipped with the negative of the usual Lie bracket, (c.f. A.Kriegl and P. W. Michor [21], Section 43.1, p. 454-456).
Here we discuss the tangential properties of Lie algebras of vector fields to an abstractsubgroup of the diffeomorphism group of a manifold. The results of this section will be ap-plied in the following to the investigation of tangent Lie algebras of the holonomy subgroupof the diffeomorphism group of an indicatrix IxM and to the fibred holonomy subgroup ofthe diffeomorphism group of the indicatrix bundle I(M).
Let M be a C∞ manifold, let G be a (not necessarily differentiable) subgroup of the diffeo-morphism group Diff∞(M) and let X∞(M) be the Lie algebra of smooth vector fields onM .
Definition 5 A vector field X∈X∞(M) is called tangent to the subgroup G of Diff∞(M),if there exists a C1-differentiable 1-parameter family {φt ∈ G}t∈(−ε,ε) of diffeomorphisms of
M such that φ0 = Id and ∂φt
∂t
∣∣t=0
= X. A Lie subalgebra g of X∞(M) is called tangent toG, if all elements of g are tangent vector fields to G.
Unfortunately, it is not true, that tangent vector fields to the group G generate a tangent Liealgebra to G. This is why we have to introduce a stronger tangential property in Definition7.
Definition 6 A C∞-differentiable k-parameter family {φ(t1,...,tk) ∈ Diff∞(M)}ti∈(−ε,ε) ofdiffeomorphisms of M is called a commutator-like family if it satisfies the equations
φ(t1,...,tk) = Id, whenever tj = 0 for some 1 ≤ j ≤ k.
We remark, that any C1-differentiable 1-parameter family {φt ∈ G}t∈(−ε,ε) of diffeomor-phisms of M with φ0 = Id is a commutator-like family. Moreover, the commutators ofcommutator-like families are commutator-like, and the inverse of commutator-like familiesare commutator-like.
Definition 7 A vector field X ∈ X∞(M) is called strongly tangent to the subgroup G ofDiff∞(M), if there exists a commutator-like family {φ(t1,...,tk) ∈ Diff∞(M)}ti∈(−ε,ε) of dif-feomorphisms satisfying the conditions
(A) φ(t1,...,tk) ∈ G for all ti ∈ (−ε, ε), 1 ≤ i ≤ k,
(B)∂kφ(t1,...,tk)
∂t1···∂tk
∣∣(0,...,0)
= X.
It follows from the commutator-like property that∂kφ(t1,...,tk)
∂t1···∂tk
∣∣(0,...,0)
is the first non-necessarily
vanishing derivative of the diffeomorphism family {φ(t1,...,tk)} at any point x ∈M , and there-fore it determines a vector field. On the other hand, by reparametrizing the commutatorlike family of diffeomorphism, it can be shown that if a vector field is strongly tangent to agroup G, then it is also tangent to G. Moreover, we have the following
Theorem 8 Let V be a set of vector fields strongly tangent to the subgroup G of Diff∞(M).The Lie subalgebra v of X∞(M) generated by V is tangent to G.
Proof. First, we investigate some properties of vector fields strongly tangent to the groupG.
Lemma 9 Let {ψ(t1,...,th) ∈ Diff∞(U)}ti∈(−ε,ε) be a C∞-differentiable h-parameter commuta-tor-like family of (local) diffeomorphisms on a neighbourhood U ⊂ Rn. Then
(i)∂i1+...+ihψ(t1,...,th)
∂ti11 ... ∂tihh
∣∣∣∣∣(0,...,0)
(x) = 0, if ip = 0 for some 1 ≤ p ≤ h;
13
(ii)∂h(ψ(t1,...,th))
−1
∂t1 ... ∂th
∣∣∣(0,...,0)
(x) = −∂hψ(t1,...,th)
∂t1 ... ∂th
∣∣∣(0,...,0)
(x);
(iii)∂hψ(t1,...,th)
∂t1 ... ∂th
∣∣∣(0,...,0)
(x) =∂ψ h√t,..., h√t)
∂t
∣∣t=0
(x)
at any point x ∈ U .
Proof. Assertions (i) and (ii) can be obtained by direct computation. It follows from (i) that∂hψ(t1,...,th)
∂t1 ... ∂th
∣∣∣(0,...,0)
(x) is the first non-necessarily vanishing derivative of the diffeomorphism
family {ψ(t1,...,th)} at any point x ∈M . Using
ψ(t1,...,tk)(x) = x+ t1 · · · tk (X(x) + ω(x, t1, . . . , tk)) ,
where limti→0
ω(x, t1, . . . , tk) = 0 we obtain, that
∂
∂t
∣∣∣t=0
ψ( k√t,..., k√t)(x) =∂
∂t
∣∣∣t=0
(x+ t
(X(x) + ω(x,
k√t, . . . ,
k√t)))
= X(x),
which proves (iii).
We remark that assertion (iii) means that any vector field strongly tangent to G is tangentto G.Now, we generalize a well-known relation between the commutator of vector fields and thecommutator of their induced flows.
Lemma 10 Let {φ(s1,...,sk)} and {ψ(t1,...,tl)} be C∞-differentiable k-parameter, respectivelyl-parameter families of (local) diffeomorphisms defined on a neighbourhood U ⊂ Rn. As-sume that φ(s1,...,sk) = Id, respectively ψ(t1,...,tl) = Id, if some of their variables is 0. Thenthe family of (local) diffeomorphisms [φ(s1,...,sk), ψ(t1,...,tl)] defined by the commutator of thegroup Diff∞(U) fulfills [φ(s1,...,sk), ψ(t1,...,tl)] = Id, if some of its variables equals 0. Moreover
∂k+l[φ(s1...sk), ψ(t1...tl)]
∂s1 ... ∂sk ∂t1 ... ∂tl
∣∣∣(0...0;0...0)
(x) = −
[∂kφ(s1...sk)
∂s1 ... ∂sk
∣∣∣(0...0)
,∂lψ(t1...tl)
∂t1 ... ∂tl
∣∣∣(0...0)
](x)
at any point x ∈ U .
Proof. The group theoretical commutator[φ(s1,...,sk), ψ(t1,...,tl)
]of the families of diffeo-
morphisms satisfies [φ(s1,...,sk), ψ(t1,...,tl)] = Id, if some of its variables equals 0. Hence
∂i1+...+ik+j1+...+jl [φ(s1,...,sk), ψ(t1,...,tl)]
∂si11 ... ∂sikk ∂tj11 ... ∂till
∣∣∣(0,...,0;0,...,0)
= 0,
if ip = 0 or jq = 0 for some index 1 ≤ p≤ k or 1≤ q ≤ l. The families of diffeomorphisms{φ(s1,...,sl)}, {ψ(t1,...,tl)}, {φ
−1(s1,...,sl)
} and {ψ−1(t1,...,tl)
} are the constant family Id, if some of
their variables equals 0. Hence one has
∂k+l[φ(s1...sk), ψ(t1...tl)]
∂s1 ... ∂sk ∂t1 ... ∂tl
∣∣∣(0,...,0; 0,...,0)
(x) = (21)
=∂k
∂s1...∂sk
∣∣∣(0...0)
(∂l(φ−1
(s1...sk)◦ ψ−1(t1...tl)
◦ φ(s1...sk)◦ ψ(t1...tl)(x))
∂t1...∂tl
∣∣∣(0...0)
)
=∂k
∂s1...∂sk
∣∣∣(0...0)
d(φ−1(s1...sk))φ(s1...sk)(x)
∂lψ−1(t1...tl)
∂t1...∂tl
∣∣∣∣∣(0,...,0)
(φ(s1...sk)(x))
,
14
where d(φ−1
(s1,...,sk)
)φ(s1,...,sk)(x)
denotes the Jacobi operator of the map φ−1(s1,...,sk) at the
point φ(s1,...,sk)(x). Using the fact, that {φ(s1,...,sk)} is the constant family Id, if some of its
variables equals 0, and the relation d(φ−1(0,...,0))φ(s1,...,sk)(x) = Id, we obtain that (21) can be
written as
d(∂kφ−1
(s1...sk)
∂s1...∂sk
∣∣∣(0...0)
)x
∂lψ−1(t1...tl)
(x)
∂t1...∂tl
∣∣∣(0...0)
+ d(∂lψ−1
(t1...tl)
∂t1...∂tl
∣∣∣(0...0)
)x
∂kφ(s1...sk)(x)
∂s1...∂sk
∣∣∣(0,...,0)
.
According to assertion (ii) of Lemma 9 the last formula gives
d(∂kφ(s1...sk)
∂s1 ... ∂sk
∣∣∣(0...0)
)x
∂lψ(t1...tl)(x)
∂t1 ... ∂tl
∣∣∣(0...0)
− d(∂lψ(t1...tl)
∂t1 ... ∂tl
∣∣∣(0...0)
)x
∂kφ(s1...sk)(x)
∂s1 ... ∂sk
∣∣∣(0...0)
,
which is the Lie bracket of vector fields[∂lψ(t1,...,tl)
∂t1 ... ∂tl
∣∣∣(0,...,0)
,∂kφ(s1,...,sk)
∂s1 ... ∂sk
∣∣∣(0,...,0)
]: U → Rn.
The previous lemma gives for 1-parameter families {φt ∈ G}t∈(−ε,ε) and {ψt ∈ G}t∈(−ε,ε) ofdiffeomorphisms of M with φ0 = ψ0 = Id the relation
∂2
∂s∂t
∣∣∣s=0,t=0
[φs, ψt] = −
[∂φs∂s
∣∣∣s=0
,∂lψt∂t
∣∣∣t=0
]. (22)
Lemma 11 Any Lie subalgebra of X∞(M) algebraically generated by strongly tangent vectorfields to the group G has a basis consisting of vector fields strongly tangent to the group G.
Proof. Let V be a set of strongly tangent vector fields to the group G and v the Liealgebra algebraically generated by V. The iterated Lie brackets of vector fields belongingto V linearly generate the vector space v. It follows from Lemma 10 that these iterated Liebrackets of vector fields are strongly tangent to the group G. Hence v is linearly generatedby vector fields strongly tangent to G.
Lemma 12 Linear combinations of vector fields tangent to G are tangent to G.
Proof. If X and Y are vector fields tangent to G then there exist C1-differentiable 1-parameter families of diffeomorphisms {φt ∈ G} and {ψt ∈ G} such that
φ0 =ψ0 = Id,∂
∂t
∣∣∣t=0
φt = X,∂
∂t
∣∣∣t=0
ψt = Y.
Considering the C1-differentiable 1-parameter families of diffeomorphisms {φt◦ψt} and {φct}one has
X + Y =∂
∂t
∣∣∣t=0
(φt ◦ ψt), cX =∂
∂t
∣∣∣t=0
φ(c t), for all c ∈ Rn,
which proves the assertion.
Lemmas 9 – 12 prove Theorem 8.
15
3.1 Diffeomorphism group of compact manifolds
The group Diff∞(K) of diffeomorphisms of a compact manifold K is an infinite dimensionalLie group belonging to the class of Frechet Lie groups. The Lie algebra of Diff∞(K) is theLie algebra X∞(K) of smooth vector fields on K endowed with the negative of the usual Liebracket of vector fields. The Frechet Lie group Diff∞(K) is modeled on the locally convextopological Frechet vector space X∞(K). A sequence {fj}j∈N ⊂ X∞(K) converges to fin the topology of X∞(K) if and only if the functions fj and all their derivatives convergeuniformly to f , respectively to the corresponding derivatives of f . We note that the difficultyof the theory of Frechet manifolds comes from the fact that the inverse function theorem andthe existence theorems of differential equations, which are well known for Banach manifolds,are not true in this category. These problems have led to the concept of regular Frechet Liegroups (cf. H. Omori [32] Chapter III, A. Kriegl – P. W. Michor [21] Chapter VIII). Thedistinguishing properties of regular Frechet Lie groups groups can be summarized as a) theexistence smooth exponential map from the Lie algebra of the Frechet Lie groups to thegroup itself, b) the existence of product integrals, which produces the convergence of someapproximation methods for solving differential equations (cf. Section III.5. in [32], pp. 83–89). J. Teichmann gave a detailed discussion of these properties in [40].
If K is a compact manifold then Diff∞(K) is a F -regular infinite dimensional Lie groupmodeled on the vector space X∞(K). Particularly Diff∞(K) is a strong ILB-Lie group.In this category of group one can define the exponential mapping and the group structureis locally determined by the Lie algebra by the exponential mapping. The Lie algebra ofDiff∞(K) is X∞(K) equipped with the negative of the usual Lie bracket (cf. [31, 32]).
Proposition 13 If a Lie subalgebra g of the Lie algebra X∞(K) of smooth vector fields on acompact manifold K is tangent to a subgroup G of the diffeomorphism group Diff∞(K) of K,then the group generated by the exponential image exp(g) of g is contained in the topologicalclosure G of G in Diff∞(K).
Proof. Let us denote by⟨
exp(g)⟩
the group generated by the exponential image of g. Forany element X ∈ g there exists a C1-differentiable 1-parameter family {Φ(t) ∈ G}t∈R ofdiffeomorphisms of the manifold K such that
Φ(0) = Id and∂Φ(s)
∂s
∣∣∣s=0
= X.
Then, considering Φ(t) as ”hair” and using the argument of Corollary 5.4. in [32], p. 85, weget that {
Φ( tn
)n}t∈R
=
{Φ( tn
)◦ · · · ◦ Φ
( tn
)}t∈R⊂ G, n = 1, 2 . . .
as a sequence of Diff∞(K) converges uniformly in all derivatives to exp(tX). It follows thatwe have {exp(tX); t ∈ R} ⊂ G for any X∈g) and therefore exp(g)⊂ G. Naturally, if for thegenerated group
⟨exp(g)
⟩, then the containing relation is preserved, that is
⟨exp(g)
⟩⊂ G,
which proves the proposition.
4 Curvature algebra
4.1 Curvature vector fields at a point
Definition 14 A vector field ξ ∈ X∞(IM) on the indicatrix bundle IM is a curvaturevector field of the Finsler manifold (M,F), if there exist vector fields X,Y ∈ X∞(M) on themanifold M such that ξ = r(X,Y ), where for every x ∈M and y ∈ IxM we have
r(X,Y )(x, y) := R(x,y)(Xx, Yx). (23)
16
If x ∈ M is fixed and X,Y ∈ TxM , then the vector field y → r(X,Y )(x, y) on IxM is acurvature vector field at the point x.
The Lie algebra R(M) of vector fields generated by the curvature vector fields of (M,F)is called the curvature algebra of the Finsler manifold (M,F). For a fixed x ∈ M theLie algebra Rx of vector fields generated by the curvature vector fields at x is called thecurvature algebra at the point x.
In this section we will investigate the properties of the curvature vector fields and of thecurvature algebra at a fixed point x∈M .Since the Finsler metric is preserved by parallel translations, its derivatives with respect tohorizontal vector fields are identically zero. Using (7) we obtain, that the derivative of theFinsler metric with respect to curvature vector fields vanishes, and hence
g(x,y)
(y,R(x,y)(lyX, lyY )
)= 0, for any y,X, Y ∈ TxM
(c.f. [34], eq. (10.9)). This means that the curvature vector fields ξ = rx(X,Y ) are tangentto the indicatrix. In the sequel we investigate the tangential properties of the curvaturealgebra to the holonomy group of the canonical connection ∇ of a Finsler manifold.
Proposition 15 Any curvature vector field at x ∈M is strongly tangent to the holonomygroup Hol(x).
Proof. Indeed, let us consider the curvature vector field ξ := rx(X,Y ) ∈ X(IxM) corre-sponding
M
v
τs,tv
ψt
φsx
Figure 7
the directions X,Y ∈TxM and letX, Y ∈ X(M) be commuting vec-tor fields i.e. [X, Y ] = 0 such thatXx=X, Yx=Y . By the geometricconstruction, the flows {φt} and{ψs} of X and Y are commuting,that is φs ◦ ψt ≡ ψt ◦ φs. For anysufficiently small s, t ∈ R we can consider the curve Ps,t defined as follows:
Ps,t(u) =
ψu(x), 0 ≤ u ≤ t,φu(ψt(x)), t ≤ u ≤ t+ s,
ψ−1u (φs(ψt(x))), t+ s ≤ u ≤ 2t+ s,
φ−1u (ψ−1
t (φs(ψt(x)))), 2t+ s ≤ u ≤ 2t+ 2s.
Because of the commuting property of the flows {φt} and {ψs} the curves Ps,t are closedparallelograms: their initial and final point at u = 0 and u = 2t + 2s are the same x ∈ M .Consequently, the parallel translation τs,t : TxM → TxM along the parallelogram Ps,t is aholonomy element for every small value of t, s ∈ R (see Figure 7).
On the other hand, using the geometric construction of parallel translation presented inSection 2.1.2, we know that the flows {φht } and {ψhs } of the horizontal lifts l(X) and l(Y ) canbe considered as parallel translations along integral curves of X and Y respectively. Theycan be considered as fiber preserving diffeomorphisms of the bundle IM for any t, s ∈ R.Then the commutator
τs,t = [φhs , ψht ] = φh−s ◦ ψh−t ◦ φhs ◦ ψht : IM → IM
is also a fiber preserving diffeomorphism of the bundle IM for any t, s ∈ R. Therefore forx ∈M the restriction
τs,t(x) = τs,t∣∣IxM
: IxM → IxM
17
to the fiber IxM is a 2-parameter C∞-differentiable family of diffeomorphisms contained inthe holonomy group Hol(x) such that
τs,0(x) = Id, τ0,t(x) = Id, and∂2
∂t∂s
∣∣∣s=0,t=0
τs,t(x) = rx(X,Y ),
which proves that the curvature vector field ξ=rx(X,Y ) is strongly tangent to the holonomygroup Hol(x) and hence we obtain the assertion.
In order to enlighten the construction and the geometric meaning of the curvature vectorfield ξ=rx(X,Y ) we consider the Figure 8 which can be seen as the extension of Figure 7.
M
TM
v
τs,tv
v
τs,tv
ψt
ψht
φs
φhs
x
ξv
Figure 8
Here we can present not only thegeometric objects at the level ofthe manifold M but at the level ofthe tangent manifold TM too. Weremark that, as it is usual, the tan-gent vectors of M are representedas ”arrows” at the level of M , butthey are represented as ”points” atthe level of the tangent space TM .For example the vectors v and τs,tvare represented as arrows at x onM and points above x in TM . Thegray vectors at the level of M rep-resent the elements of the parallelvector field V along the parallel-ogram Ps,t with the initial condi-tion Vx = v. The gray dots are the points in TM corresponding to the elements of V .These dots lie on the curves of the flows φht , ψhs because V is a parallel field and these flowscorresponds to the parallel translations along integral curves of X and Y respectively. Asthe picture shows, the parallel translation along the parallelogram Ps,t of a vector v ∈ TxMcan be obtained by following in TM the flows φht , ψhs , φh−t, ψ
h−s above Ps,t. The indicatrix,
or unite ball, at x ∈ M is represented by the oval above x. Since the parallel translationpreserves the norm, if v ∈ IxM , then τs,tv ∈ IxM . Therefore t → 1
2τt,t(v) is a curve inIxM . Its tangent vector at t = 0 is ξ(v) which is therefore a tangent vector of IxM .
M
TM
v
v
ψt
ψht
φs
φhs
x
Ip
Figure 9
We remark that in the case,when the curvature is identicallyzero, the horizontal lifts of com-muting vector fields are also com-muting vector fields. Thereforeone obtain φhs ◦ ψht ≡ ψht ◦ φhs andτs,t = [φhs , ψ
ht ] : IM → IM is the
identity transformation. In thatcase τs,tv ≡ v is a constant mapand therefore its derivative is zero,that is ξv = 0. Geometrically thatmeans that the horizontal lifts ofthe closed parallelograms Ps,t areclosed parallelograms. See Figure9.
Theorem 16 The curvature algebra Rx at a point x∈M of a Finsler manifold (M,F) hasthe following properties:
(i) Rx is tangent to the holonomy group Hol(x),
18
(ii) the group generated by the exponential image exp(Rx
)is a subgroup of the topological
closure of the holonomy group Hol(x).
Proof. Since by Proposition 15 the curvature vector fields are strongly tangent to Hol(x)and the curvature algebra Rx is algebraically generated by the curvature vector fields, theassertion (i) follows from Theorem 8. Assertion (ii) is a consequence of Proposition 13.
Proposition 17 The curvature algebra Rx of a Riemannian manifold (M, g) at any pointx ∈ M is isomorphic to the linear Lie algebra over the vector space TxM generated by thecurvature operators of (M, g) at x ∈M .
Proof. The curvature tensor field of a Riemannian manifold given by the equation (7) islinear with respect to y ∈ TxM and hence
R(x,y)(ξ, η) = (Rx(ξ, η))kl yl ∂
∂yk,
where Rx(ξ, η))kl is the matrix of the curvature operator Rx(ξ, η) : TxM → TxM with re-spect to the natural basis
{∂∂x1 |x, ..., ∂
∂xn |x}. Hence any curvature vector field rx(ξ, η)(y)
with ξ, η ∈ TxM has the shape rx(ξ, η)(y) = (Rx(ξ, η))kl yl ∂∂yk
. It follows that the flow of
rx(ξ, η)(y) on the indicatrix IxM generated by the vector field rx(ξ, η)(y) is induced by theaction of the linear 1-parameter group exp tRx(ξ, η)) on TxM , which implies the assertion.
Since for Finsler surfaces of non-vanishing curvature the curvature vector fields form a one-dimensional vector space and hence the generated Lie algebra is also one-dimensional, wehave
Remark 18 The curvature algebra of Finsler surfaces is at most one-dimensional.
4.2 Constant curvature
Now, we consider a Finsler manifold (M,F) of non-zero constant curvature. In this case forany x ∈M the curvature vector field rx(X,Y )(y) has the shape (cf. (14))
r(X,Y )(y) = c(δijgkm(y)ym − δikgjm(y)ym
)XjY k
∂
∂yi, 0 6= c ∈ R.
Putting yj = gjm(y)ym we can write r(X,Y )(y) = c(δijyk − δikyj
)XjY k ∂
∂yi . Any linearcombination of curvature vector fields has the form
r(A)(y) = Ajk(δijyk − δikyj
) ∂
∂yi,
where A = Ajk ∂∂xj ∧ ∂
∂xk ∈ TxM ∧ TxM is arbitrary bivector at x ∈M .
Lemma 19 Let (M,F) be a Finsler manifold of non-zero constant curvature. The curvaturealgebra Rx at any point x ∈M satisfies
dimRx ≥n(n− 1)
2, (24)
where n = dimM .
Proof. Let us consider the curvature vector fields rjk = rx( ∂∂yj ,
∂∂yk
)(y) at a fixed pointx ∈M . If a linear combination
Ajkrjk = Ajk(δijyk − δikyj)∂
∂yi= (Aikyk −Ajiyj)
∂
∂yi= 2Aikyk
∂
∂yi
19
of curvature vector fields rjk with constant coefficients Ajk = −Akj satisfies Ajkrjk = 0 forany y ∈ TxM then one has the linear equation Aikyk = 0 for any fixed index i. Since thecovector fields y1, . . . , yn are linearly independent we obtain Ajk = 0 for all j, k ∈ {1, . . . , n}.It follows that the curvature vector fields rjk are linearly independent for any j < k and
hence dimRx ≥ n(n−1)2 .
Corollary 20 Let (M, g) be a Riemannian manifold of non-zero constant curvature withn = dimM . The curvature algebra Rx at any point x ∈ M is isomorphic to the orthogonalLie algebra o(n).
Proof. The holonomy group of a Riemannian manifold is a subgroup of the orthogonalgroup O(n) of the tangent space TxM and hence the curvature algebra Rx is a subalgebraof the orthogonal Lie algebra o(n). Hence the previous assertion implies the corollary.
Theorem 21 Let (M,F) be a Finsler manifold of non-zero constant curvature with n =dimM > 2. If the point x ∈M is not (semi-)Riemannian then the curvature algebra Rx atx ∈M satisfies
dimRx >n(n− 1)
2. (25)
Proof. We assume dimRx = n(n−1)2 . For any constant skew-symmetric matrices {Ajk}
and {Bjk} the Lie bracket of vector fields Aikyk∂∂yi and Bikyk
∂∂yi has the shape Cikyk
∂∂yi ,
where {Cik} is a constant skew-symmetric matrix, too. Using the homogeneity of ghl weobtain
∂yh∂ym
=∂ghl∂ym
yl + ghm = ghm (26)
and hence [Amk yk
∂
∂ym, Bih yh
∂
∂yi
]=
(Amk Bih
∂yh∂ym
−Bmk Aih ∂yh∂ym
)yk
∂
∂yi
=(Bih ghmA
mk −Aih ghmBmk)yk
∂
∂yi= Cik yk
∂
∂yi.
Particularly, for the skew-symmetric matrices Eijab = δiaδjb−δibδja, a, b ∈ {1, . . . , n}, we have[
Eijab yj∂
∂yi, Eklcd yl
∂
∂yk
]=(Eihcd ghmE
mkab − Eihab ghmEmkcd
)yk
∂
∂yi= Λimab,cd ym
∂
∂yi,
where the constants Λijab,cd satisfy Λijab,cd =−Λjiab,cd =−Λijba,cd =−Λijab,dc =−Λijcd,ab. Puttingi = a and computing the trace for these indices we obtain
(n− 2)(gbd yc − gbc yd) = Λlb,cd yl, (27)
where Λlb,cd := Λilib,cd. The right hand side is a linear form in variables y1, . . . , yn. According
to the identity (27) this linear form vanishes for yc = yd = 0, hence Λlb,cd = 0 for l 6= c, d.
Denoting λ(c)bd := 1
n−2Λcb,cd (no summation for the index c) we get the identities
gbd yc − gbc yd = λ(c)bd yc − λ
(d)bc yd (no summation for c and d).
Putting yd = 0 we obtain gbd∣∣yd=0
= λ(c)bd for any c 6= d. It follows λ
(c)bd is independent of the
index c (6= d). Defining λbd := λ(c)bd with some c ( 6= d) we obtain from (27) the identity
gbd yc − gbc yd = λbd yc − λbc yd (28)
20
for any b, c, d ∈ {1, . . . , n}. We have
λcd yb − λcb yd = (gbd yc − gbc yd)− (gdb yc − gdc yb) = (λbd yc − λbc yd)− (λdb yc − λdc yb).
which implies the identity
(λcd yb − λcb yd) + (λdb yc − λdc yb) + (λbc yd − λbd yc) =
= (λcd − λdc) yb + (λdb − λbd) yc + (λbc − λcb) yd = 0. (29)
Since dimM > 2, we can consider 3 different indices b, c, d and we obtain from the identity(29) that λbc = λcb for any b, c ∈ {1, . . . , n}.
By derivation the identity (28) we get
∂gbd∂ya
yc −∂gbc∂ya
yd + gbd δac − gbc δad = λbd δ
ac − λbc δad .
Using (26) we obtain
∂ya∂yq
(∂gbd∂ya
yc −∂gbc∂ya
yd
)+ gbd gcq − gbc gdq =
=∂gbd∂yq
yc −∂gbc∂yq
yd + gbd gcq − gbc gdq = λbd gcq − λbc gdq.
Since (∂gbd∂yq
yc −∂gbc∂yq
yd
)yb = 0
we get the identityyd gcq − yc gdq = λbd y
b gcq − λbc yb gdq.
Multiplying both sides of this identity by the inverse {gqr} of the matrix {gcq} and takingthe trace with respect to the indices c, r we obtain the identity
(n− 1) yd = (n− 1)λbd yb.
Hence we obtain that gbd yb = λbd y
b and hence gbd = λbd, which means that the pointx ∈M is (semi-)Riemannian. From this contradiction follows the assertion.
Corollary 22 The curvature algebra Rx at a point x ∈M of a Finsler manifold (M,F) ofnon-zero constant curvature satisfies
dimRx =n(n− 1)
2, where n=dimM, (30)
if and only if n= 2 or the point x ∈M is (semi-)Riemannian.
Theorem 23 Let (M,F) be a positive definite Finsler manifold of non-zero constant cur-vature with n = dimM > 2. The holonomy group of (M,F) is a compact Lie group if andonly if (M,F) is a Riemannian manifold.
Proof. We assume that the holonomy group of a Finsler manifold (M,F) of non-zeroconstant curvature with dimM ≥ 3 is a compact Lie transformation group on the indicatrixIxM . The curvature algebra Rx at a point x ∈M is tangent to the holonomy group Hol(x)and hence dim Hol(x) ≥ dimRx. If there exists a not (semi-)Riemannian point x ∈M then
dimRx >n(n−1)
2 . The (n − 1)-dimensional indicatrix IxM at x can be equipped with aRiemannian metric which is invariant with respect to the compact Lie transformation groupHol(x). Since the group of isometries of an n − 1-dimensional Riemannian manifold is of
21
dimension at most n(n−1)2 (cf. Kobayashi [17], p. 46,) we obtain a contradiction, which
proves the assertion.
Since the holonomy group of a Landsberg manifold is a subgroup of the isometry group ofthe indicatrix, we obtain that any Landsberg manifold of non-zero constant curvature withdimension > 2 is Riemannian (c.f. Numata [30]).
We can summarize our results as follows:
Theorem 24 The holonomy group of any non-Riemannian positive definite Finsler man-ifold of non-zero constant curvature with dimension > 2 does not occur as the holonomygroup of any Riemannian manifold.
4.3 Infinite dimensional curvature algebra
Let us consider the singular (non y-global) Finsler manifold (H3,F), where H3 is the 3-dimensional Heisenberg group and F is a left-invariant Berwald-Moor metric (c.f. [34],Example 1.1.5, p. 8).
The group H3 can be realized as the Lie group of matrices of the form
[1 x1 x2
0 1 x3
0 0 1
], where
x = (x1, x2, x3) ∈ R3 and hence the multiplication can be written as
(x1, x2, x3) · (y1, y2, y3) = (x1 + y1, x2 + y2 + x1y3, x3 + y3).
The vector 0 = (0, 0, 0) ∈ R3 gives the unit element of H3. The Lie algebra h3 = T0H3
consists of matrices of the form
[0 a1 a2
0 0 a3
0 0 0
], corresponding to the tangent vector a=a1 ∂
∂x1 +
a2 ∂∂x2 + a3 ∂
∂x3 at the unit element 0 ∈ H3. A left-invariant Berwald-Moor Finsler metric Fis induced by the (singular) Minkowski functional F
0: h3 → R:
F0(a) :=
(a1a2a3
) 23
of the Lie algebra in the following way: if y = (y1, y2, y3) is a tangent vector at x ∈ H3,then
F(x, y) := F0(x−1y).
The coordinate expression of the singular (non y-global) Finsler metric F is
F(x, y) =(y1(y2−x1y3
)y3) 2
3 .
Since F is left-invariant, the associated geometric structures (connection, geodesics, curva-ture) are also left-invariant and the curvature algebras at different points are isomorphic.Using the notation
rx(i, j) = rx
( ∂
∂xi,∂
∂xj
), i, j = 1, 2, 3,
for curvature vector fields, a direct computation yields
rx(1, 2) =1
4
(5y1
2y3
2
(x1y3−y2)3∂
∂y1+y1y3
2 (3x1y3 + y2
)(y2−x1y3)3
∂
∂y2+
4y1y33
(y2−x1y3)3∂
∂y3
),
rx(1, 3) =1
4
(y1
2y3(6x1y3−11 y2
)(x1y3−y2)3
∂
∂y1+
4y1y32x1(2x1y3−3 y2
)(y2−x1y3)3
∂
∂y2+
+y1y3
2 (7x1y3−11 y2
)(y2−x1y3)3
∂
∂y3
),
rx(2, 3) =1
4
(4y1
3y3
(x1y3−y2)3∂
∂y1+y1
2y3(6x1y3−y2
)(y2−x1y3)3
∂
∂y2+
5y12y3
2
(y2−x1y3)3∂
∂y3
).
22
The curvature vector fields r0(i, j), i, j = 1, 2, 3, at the unit element 0 ∈ H3 generate the
curvature algebra r0. Let us denote Y k,m := y1ky3
m
y2k+m−1 , k,m ∈ N, and consider the vector
fields
Ak,m(a1, a2, a3) = a1Y k+1,m ∂
∂y1
∣∣∣0
+ a2Y k,m∂
∂y2
∣∣∣0
+ a3Y k,m+1 ∂
∂y3
∣∣∣0, (31)
with (a1, a2, a3) ∈ R3 and k,m ∈ N. Then the curvature vector fields r0(i, j) at 0 ∈ H3 canbe written in the form
r0(1, 2) =1
4A1,2(−5, 1, 4), r0(1, 3) =
1
4A1,1(11, 0,−11), r0(2, 3) =
1
4A2,1(−4,−1, 5).
Proposition 25 The curvature algebra rx at any point x ∈ M is a Lie algebra of infinitedimension.
Proof. Since the Finsler metric is left-invariant, the curvature algebras at different pointsare isomorphic. Therefore it is enough to prove that the curvature algebra r0 at 0∈H3 hasinfinite dimension. We prove the statement by contradiction: let us suppose that r0 is finitedimensional.A direct computation shows that for any (a1, a2, a3), (b1, b2, b3) ∈ R3 one has[
Ak,m(a1, a2, a3), Ap,q(b1, b2, b3)]
= Ak+p,m+q(c1, c2, c3)
with some (c1, c2, c3) ∈ R3. It follows that any iterated Lie bracket of curvature vector fieldsr0(i, j), i, j = 1, 2, 3, has the shape (31) and hence there exists a basis of the curvaturealgebra r0 of the form {Aki,mi(a1
i , a2i , a
3i )}Ni=1, where N ∈ N is the dimension of r0. We can
assume that {(ki,mi)}Ni=1 forms an increasing sequence, i.e. (k1,m1) ≤ (k2,m2) ≤ · · · ≤(kN ,mN ) holds with respect to the lexicographical ordering of N× N. We can consider thevector fields
4
11r0(1, 3) = A1,1(1, 0,−1), 4r0(1, 2) = A1,2(−5, 1, 4), 4r0(2, 3) = A2,1(−4,−1, 5)
as the first three members of this sequence. Hence 1 ≤ kN ,mN and[A1,1(1, 0,−1), AkN ,mN (a1
N , a2N , a
3N )]
= A1+kN ,1+mN (c1, c2, c3)
belongs to r0, too, where c1 = (kN − mN − 1)a1N + 2a2
N − a3N , c2 = (kN − mN )a2
N andc3 = a1
N−2a2N+(kN−mN+1)a3
N . Since kN < 1+kN , mN < 1+mN we have c1 = c2 = c3 = 0and hence the homogeneous linear system
0 = (kN −mN − 1)a1N + 2a2
N − a3N ,
0 = (kN −mN )a2N ,
0 = a1N − 2a2
N + (kN −mN + 1)a3N
has a solution (a1N , a
2N , a
3N ) 6= (0, 0, 0). It follows that kN = mN .
Similarly, computing the Lie bracket
0 =[A1,2(−5, 1, 4), AkN ,kN (a1
N , a2N , a
3N )]
= A1+kN ,2+kN (d1, d2, d3)
Since kN < 1 + kN < 2 + kN we have d1 = d2 = d3 = 0 giving the homogeneous linearsystem
0 =(−3kN + 5)a1N − 15a2
N + 10a3N ,
0 =− a1N + (3− 3kN )a2
N − 2a3N ,
0 =− 4a1N + 12a2
N − (3kN + 8)a3N
for (a1N , a
2N , a
3N ). The determinant of this system vanishes only for kN = 0 which is a
contradiction.
23
Corollary 26 The holonomy group of the Finsler manifold (H3,F) has an infinite dimen-sional tangent Lie algebra.
We remark here, that it remains an interesting open question: Is there a nonsingular (y-global) Finsler manifold whose curvature algebra is infinite dimensional?
5 Holonomy algebra
5.1 Fibred holonomy group
Now, we introduce the notion of the fibred holonomy group of a Finsler manifold (M,F) asa subgroup of the diffeomorphism group of the total manifold IM of the bundle (IM,π,M)and apply our results on tangent vector fields to an abstract subgroup of the diffeomorphismgroup to the study of tangent Lie algebras to the fibred holonomy group.
Definition 27 The fibred holonomy group Holf(M) of (M,F) consists of fibre preservingdiffeomorphisms Φ ∈ Diff∞(IM) of the indicatrix bundle (IM,π,M) such that for anyp ∈M the restriction Φp = Φ|IpM ∈ Diff∞(IpM) belongs to the holonomy group Hol(p).
We note that the holonomy group Hol(p) and the fibred holonomy group Holf(M) are topo-logical subgroups of the infinite dimensional Lie groups Diff∞(IpM) and Diff∞(IM) respec-tively.
The definition of strongly tangent vector fields yields
Remark 28 A vector field ξ ∈ X∞(IM) is strongly tangent to the fibred holonomy groupHolf(M) if and only if there exists a family
{Φ(t1,...,tk)
∣∣IM
}ti∈(−ε,ε) of fibre preserving dif-
feomorphisms of the bundle (IM,π,M) such that for any indicatrix Ip the induced family{Φ(t1,...,tk)
∣∣IpM
}ti∈(−ε,ε) of diffeomorphisms is contained in the holonomy group Hol(p) and
ξ∣∣IpM
is strongly tangent to Hol(p).
Since π(Φ(t1,...,tk)(p)
)≡p and π∗(ξ)=0 for every p ∈ U , we get the
Corollary 29 Strongly tangent vector fields to the fibred holonomy group Holf(M) are verti-cal vector fields. If ξ ∈ X∞(IM) is strongly tangent to Holf(M) then its restriction ξp := ξ
∣∣Ip
to any indicatrix Ip is strongly tangent to the holonomy group Hol(p).
Now we prove that the first assertion of Theorem 16 on the tangential property to theholonomy group of curvature vector fields at a point can be extended to curvature vectorfields defined on the full the indicatrix bundle.
Proposition 30 If the Finsler manifold (M,F) is diffeomorphic to Rn then any curvaturevector field ξ ∈ X∞(IM) of (M,F) on the indicatrix bundle is strongly tangent to the fibredholonomy group Holf(M).
Proof. Since M is diffeomorphic to Rn we can identify the manifold M with the vector spaceRn. Let ξ = r(X,Y ) ∈ X∞(IRn) be a curvature vector field with X,Y ∈ X∞(Rn). Accord-ing to Proposition 15 its restriction ξ
∣∣IpRn to any indicatrix IpRn is strongly tangent to the
holonomy groups Hol(p). We have to prove that there exists a family{
Φ(t1,...,tk)
∣∣IRn
}ti∈(−ε,ε)
of fibre preserving diffeomorphisms of the indicatrix bundle (IRn, π,Rn) such that for anyp ∈ Rn the family of diffeomorphisms induced on the indicatrix Ip is contained in Hol(p)and ξ
∣∣IpRn is strongly tangent to Hol(p).
For any p ∈ Rn and −1 < s, t < 1 let Π(sXp, tYp) be the parallelogram in Rn determinedby the vertexes p, p+ sXp, p+ sXp + tYp, p+ tYp ∈ Rn and let τΠ(sXp,tYp) : Ip → Ip denotethe (nonlinear) parallel translation of the indicatrix Ip along the parallelogram Π(sXp, tYp)
24
with respect to the associated homogeneous (nonlinear) parallel translation of the Finslermanifold (Rn,F). Clearly we have τΠ(sXp,tYp) = IdIRn , if s = 0 or t = 0 and
∂2τΠ(sXp,tYp)
∂s∂t
∣∣∣(s,t)=(0,0)
= ξp for every p ∈ Rn.
Since Π(sXp, tYp) is a differentiable field of parallelograms in Rn, the maps τΠ(sXp,tYp),p ∈ Rn, 0 < s, t < 1, define a 2-parameter family of fibre preserving diffeomorphisms of theindicatrix bundle IRn. The diffeomorphisms induced by the family
{τΠ(sXp,tYp)
}s,t∈(−1,1)
on any indicatrix Ip are contained in Hol(p). Hence the vector field ξ∈X∞(Rn) is stronglytangent to the fibred holonomy group Holf(M), hence the assertion is proved.
Corollary 31 If M is diffeomorphic to Rn then the curvature algebra R(M) of (M,F) istangent to the fibred holonomy group Holf(M).
5.2 Infinitesimal holonomy algebra
The following assertion shows that similarly to the Riemannian case, the curvature algebracan be extended to a larger tangent Lie algebra containing all horizontal Berwald covariantderivatives of the curvature vector fields.
Proposition 32 If ξ ∈ X∞(IM) is strongly tangent to the fibred holonomy group Holf(M)of (M,F), then its horizontal Berwald covariant derivative ∇Xξ along any vector field X ∈X∞(M) is also strongly tangent to Holf(M).
Proof. Let τ be the (nonlinear) parallel translation along the flow ϕ of the vector field X, i.e.for every p ∈M and t ∈ (−εp, εp) the map τt(p) : IpM → Iϕt(p)M is the (nonlinear) paralleltranslation along the integral curve of X. If {Φ(t1,...,tk)}ti∈(−ε,ε) is a C∞-differentiable k-parameter family {Φ(t1,...,tk)}ti∈(−ε,ε) of fibre preserving diffeomorphisms of the indicatrixbundle (IM,π|M ,M) satisfying the conditions of Definition 29 then the commutator
[Φ(t1,...,tk), τtk+1] := Φ−1
(t1,...,tk) ◦ (τtk+1)−1 ◦ Φ(t1,...,tk) ◦ τtk+1
of the group Diff∞(IM
)fulfills [Φ(t1,...,tk), τtk+1
] = Id, if some of its variables equals 0.Moreover
∂k+1[Φ(t1...tk), τ(tk+1)]
∂t1 ... ∂tk+1
∣∣∣∣∣(0...0)
= −[ξ,Xh
](32)
at any point of M , which shows that the vector field[ξ,Xh
]is strongly tangent to Holf(M).
Moreover, since the vector field ξ is vertical, we have h[Xh, ξ] = 0, and using ∇Xξ := [Xh, ξ]we obtain
−[ξ,Xh] = [Xh, ξ] = v[Xh, ξ] = ∇Xξ
which yields the assertion.
Definition 33 Let hol∗(M) be the smallest Lie algebra of vector fields on the indicatrixbundle IM satisfying the properties
(i) any curvature vector field ξ belongs to hol∗(M),
(ii) if ξ, η ∈ hol∗(M) then [ξ, η] ∈ hol∗(M),
(iii) if ξ ∈ hol∗(M) and X∈X∞(M) then the horizontal Berwald covariant derivative ∇Xξalso belongs to hol∗(M).
The Lie algebra hol∗(M) ⊂ X∞(IM) is called the infinitesimal holonomy algebra of theFinsler manifold (M,F).
25
Remark 34 The infinitesimal holonomy algebra hol∗(M) is invariant with respect to thehorizontal Berwald covariant derivation, i.e.
ξ ∈ hol∗(M) and X ∈ X∞(M) ⇒ ∇Xξ ∈ hol∗(M). (33)
The results of this sections yield the following
Theorem 35 If M is diffeomorphic to Rn then the infinitesimal holonomy algebra hol∗(M)is tangent to the fibred holonomy group Holf(M).
Let hol∗(M) ⊂ X∞(IM) be the infinitesimal holonomy algebra of the Finsler manifold(M,F) and let p be a a given point in M .
Definition 36 The Lie algebra hol∗(p) :={ξp ; ξ ∈ hol∗(M)
}⊂ X∞(IpM) of vector fields
on the indicatrix IpM is called the infinitesimal holonomy algebra at the point p ∈M .
Clearly, for any p ∈M the curvature algebra Rp at p ∈M is contained in the infinitesimalholonomy algebra hol∗(p) at p ∈M .The following assertion is a direct consequence of the definition. It shows that the infinites-imal holonomy algebra at a point p of (M,F) can be calculated in a neighbourhood ofp.
Remark 37 Let (U,F|U ) be an open submanifold of (M,F) such that U ⊂M is diffeomor-phic to Rn and let p ∈ U . The infinitesimal holonomy algebras at p of the Finsler manifolds(M,F) and (U,F|U ) coincide.
Now, we can prove the following
Theorem 38 The infinitesimal holonomy algebra hol∗(p) at a point p∈M has the followingproperties:
(i) hol∗(p) is tangent to the holonomy group Hol(p),
(ii) the group generated by the exponential image exp(hol∗(p)
)is a subgroup of the topo-
logical closure of the holonomy group Hol(p).
Proof. Let U ⊂ M be an open submanifold of M , diffeomorphic to Rn and containingp ∈M . According to the previous remark we have hol∗(p) :=
{ξp ; ξ ∈ holf(U)
}. Since the
fibred holonomy algebra holf(U) is tangent to the fibred holonomy group Holf(U) we obtainassertion (i). Assertion (ii) is a consequence of Proposition 13.
5.3 Holonomy algebra
Let x(t), 0 ≤ t ≤ a be a smooth curve joining the points q=x(0) and p=x(a) in the Finslermanifold (M,F). If y(t) = τty(0) ∈ Ix(t)M is a parallel vector field along x(t), 0≤ t≤ a,where τt : IqM → Ix(t)M denotes the homogeneous (nonlinear) parallel translation, then
we have Dxy(t) :=(dyi(t)dt +Gij(x(t), y(t))xj(t)
)∂∂xi = 0. Considering a vector field ξ on the
indicatrix IqM , the map τa∗ξ◦τ−1a : (p, y) 7→ τa∗ξ(y(a)) gives a vector field on the indicatrix
IpM . Hence we can formulate
Lemma 39 For any vector field ξ ∈ hol∗(q) ⊂ X∞(IqM) in the infinitesimal holonomyalgebra at q the vector field τa∗ξ ◦ τ−1
a ∈X∞(IpM) is tangent to the holonomy group Hol(p).
Proof. Let {φt ∈ Hol(q)}t∈(−ε,ε) be a C1-differentiable 1-parameter family of diffeomor-phisms of IqM belonging to the holonomy group Hol(q) and satisfying the conditions φ0 = Id,∂φt
∂t
∣∣t=0
= ξ. Since the 1-parameter family
τa ◦ φt ◦ τ−1a ∈ Diff∞(IpM)}t∈(−ε,ε)
26
of diffeomorphisms consists of elements of the holonomy group Hol(p) and satisfies the con-ditions
τa ◦ φ0 ◦ τ−1a = Id,
∂(τa ◦ φt ◦ τ−1
a
)∂t
∣∣∣t=0
= τa∗ξ ◦ τ−1a ,
the assertion follows.
Definition 40 A vector field Bγξ ∈ X∞(IpM) on the indicatrix IpM will be called theBerwald translate of the vector field ξ ∈ X∞(IqM) along the curve γ = x(t) if
Bγξ = τa∗ξ ◦ (τa)−1.
Remark 41 Let y(t) = τty(0) ∈ Ix(t)M be a parallel vector field along γ=x(t), 0 ≤ t ≤ a,started at y(0) ∈ Ix(0)M . Then, the vertical vector field ξt = ξ(x(t), y(t)) along (x(t), y(t))is the Berwald translate ξt = τt∗ξ0 ◦ τt−1 if and only if
∇xξ =
(∂ξi(x, y)
∂xj−Gkj (x, y)
∂ξi(x, y)
∂yk+Gijk(x, y)ξk(x, y)
)xj
∂
∂yi= 0.
Now, lemma 39 yields the following
Corollary 42 If ξ ∈ hol∗(q) then its Berwald translate Bγξ ∈ X∞(IpM) along any curveγ=x(t), 0 ≤ t ≤ a, joining q=x(0) with p=x(a) is tangent to the holonomy group Hol(p).
This last statement motivates the following
Definition 43 The holonomy algebra holp(M) of the Finsler manifold (M,F) at the pointp ∈M is defined by the smallest Lie algebra of vector fields on the indicatrix IpM , containingthe Berwald translates of all infinitesimal holonomy algebras along arbitrary curves x(t),0 ≤ t ≤ a joining any points q = x(0) with the point p = x(a).
Clearly, the holonomy algebras at different points of the Finsler manifold (M,F) are iso-morphic. Lemma 39, Corollary 42 and Proposition 13 yield the following
Theorem 44 The holonomy algebra holp(M) at a point p ∈M of a Finsler manifold (M,F)has the following properties:
(i) holp(M) is tangent to the holonomy group Hol(p),
(ii) the group generated by the exponential image exp(holp(M)
)is a subgroup of the topo-
logical closure of the holonomy group Hol(p).
5.4 Finsler surfaces with hol∗(x) = Rx
The relation between the infinitesimal holonomy algebra and the curvature algebra is en-lightened by the following
Theorem 45 Let (M,F) be a Finsler surface with non-zero constant flag curvature. Theinfinitesimal holonomy algebra hol∗(x) at a point x ∈M coincides with the curvature algebraRx at x if and only if the mean Berwald curvature E(x,y) of (M,F) vanishes for any y ∈IxM .
Proof. Let U ⊂M be a neighbourhood of x∈M diffeomorphic to R2. Identifying U withR2 and considering a coordinate system (x1, x2) in R2 we can write
Rijk(x, y) = λ(δijgkm(x, y)ym − δikgjm(x, y)ym
), with λ 6= 0.
27
Since the curvature tensor field is skew-symmetric, R(x,y) acts on the one-dimensional wedgeproduct TxM∧TxM . According to Lemma 2 the covariant derivative of the curvature vectorfield ξ=R(X,Y )= 1
2R(X ⊗ Y −Y ⊗X)=R(X ∧ Y ) can be written in the form
∇Zξ = ∇Z (r(X,Y )) = R (∇Z(X ∧ Y )) = R(∇ZX ∧ Y +X ∧∇ZY ),
where X,Y, Z ∈ X(U). If X = Xi ∂∂xi , Y = Y i ∂
∂xi and Z = Zi ∂∂xi then we have X ∧ Y =
12
(X1Y 2 −X2Y 1
)∂∂x1 ∧ ∂
∂x2 and hence we obtain
∇Zξ = R
(∇k(
(X1Y 2 − Y 1X2)∂
∂x1∧ ∂
∂x2
)Zk)
= (34)
= R
(∂(X1Y 2 − Y 1X2)
∂xkZk
∂
∂x1∧ ∂
∂x2
)+ (X1Y 2 − Y 1X2)R
(∇k(
∂
∂x1∧ ∂
∂x2
))Zk,
where we denote the covariant derivative ∇Z by ∇k if Z = ∂∂xk , k= 1, 2. For given vector
fields X,Y, Z ∈ X∞(U) the expression ∂(X1Y 2−Y 1X2)∂xk Zk is a function on U . Hence there
exists a function ψ on U such that
R
(∂(XjY h − Y jXh)
∂xkZk
∂
∂xj∧ ∂
∂xh
)= ψR(X ∧ Y ) = ψR(X,Y ),
and ψR(X,Y ) is an element of the curvature algebra R(U) of the submanifold (U,F|U ).
Now, we investigate the second term of the right hand side of (34).
∇k( ∂
∂x1∧ ∂
∂x2
)=(∇k
∂
∂x1
)∧ ∂
∂x2+
∂
∂x1∧(∇k
∂
∂x2
)=
= Glk1
∂
∂xl∧ ∂
∂x2+
∂
∂x1∧Gmk2
∂
∂xm=(G1k1 +G2
k2
) ∂
∂x1∧ ∂
∂x2.
Hence
(X1Y 2−Y 1X2)R(∇k( ∂
∂x1∧ ∂
∂x2
))Zk=
(G1k1+G2
k2
)ZkR(X,Y )=
(G1k1 +G2
k2
)Zkξ
This expression belongs to the curvature algebra if and only if the function G1k1 +G2
k2 doesnot depend on the variable y, i.e. if and only if
Ekh =∂(G1k1 +G2
k2
)∂yh
= 0, h, k = 1, 2,
identically.
Remark 46 Let ξ = R(X,Y ) be a curvature vector field. Assume that the vector fieldsX,Y ∈X∞(M) have constant coordinate functions in a local coordinate system (x1, ..., xn)of the Finsler surface (M,F). Then we have in this coordinate system
∇Zξ =(G1k1 +G2
k2
)Zkξ.
A Finsler manifold (M,F) is called Randers manifold if its Finsler function has the formF = α+β, where α =
√αjk(x)yjyk is a Riemannian metric and β = βj(x)yj is a linear form.
Z. Shen constructed in [35] families of Randers surfaces depending on the real parameterε, which are of constant flag curvature 1 on the unit sphere S2 ⊂ R3 and of constant flagcurvature −1 on a disk D2 ⊂ R2. These Finsler surfaces are not projectively flat and havevanishing S-curvature (c.f. [35], Theorems 1.1 and 1.2). Their Finsler function is defined by
α =
√ε2h(v, y)2 + h(y, y) (1− ε2h(v, v))
1− ε2h(v, v), β =
εh(v, y)
1− ε2h(v, v), (35)
where h(v, y) is the standard metric of the sphere S2, respectively h(v, y) is the standardKlein metric on the unit disk D2 and v denotes the vector field defined by (−x2, x1, 0) at(x1, x2, x3) ∈ S2, respectively by (−x2, x1) at (x1, x2) ∈ D2.
28
Theorem 47 For any Randers surface defined by (35) the infinitesimal holonomy algebrahol∗(x) at a point x ∈M coincides with the curvature algebra Rx.
Proof. According to Theorem 1.1 and 1.2 in [35], the above classes of not locally projectivelyflat Randers surfaces with non-zero constant flag curvature have vanishing S-curvature.Moreover, Proposition 6.1.3 in [34], p. 80, states that the mean Berwald curvature vanishesif and only if the S-curvature is a linear form on the surface. Hence the assertion followsfrom Corollary 45.
6 Infinite dimensional infinitesimal holonomy algebra
6.1 Projective Finsler surfaces of constant curvature
A Finsler manifold (M,F) of dimension 2 is called Finsler surface. In this case the indicatrixis 1-dimensional at any point x ∈ M , hence the curvature vector fields at x ∈ M areproportional to any given non-vanishing curvature vector field. It follows that the curvaturealgebra Rx(M) has a simple structure: it is at most 1-dimensional and commutative. Evenin this case, the infinitesimal holonomy algebra hol∗x(M) can be higher dimensional, orpotentially infinite dimensional. For the investigation of such examples we use a classicalresult of S. Lie claiming that the dimension of a finite-dimensional Lie algebra of vector fieldson a connected 1-dimensional manifold is less than 4 (cf. [1], Theorem 4.3.4). We obtain thefollowing
Lemma 48 If the infinitesimal holonomy algebra hol∗x(M) of a Finsler surface (M,F) con-tains 4 simultaneously non-vanishing R-linearly independent vector fields, then hol∗x(M) isinfinite dimensional.
Proof. If the infinitesimal holonomy algebra is finite-dimensional, then the dimension ofthe corresponding Lie group acting locally effectively on the 1-dimensional indicatrix wouldbe at least 4, which is a contradiction.
Let (M,F) be a locally projectively flat Finsler surface of non-zero constant curvature, let(x1, x2) be a local coordinate system centered at x ∈ M , corresponding to the canonicalcoordinates of the Euclidean space which is projectively related to (M,F) and let (y1, y2)be the induced coordinate system in the tangent plane TxM .
In the sequel we identify the tangent plane TxM with R2 with help of the coordinatesystem (y1, y2). We will use the euclidean norm ||(y1, y2)|| =
√(y1)2 + (y2)2 of R2 and the
corresponding polar coordinate system (er, t), too.Let ϕ(y1, y2) be a positively 1-homogeneous function on R2 and let r(t) be the 2π-periodic
smooth function r : R→ R determined by
ϕ(er(t)cos t, er(t)sin t) = 1 or ϕ(y1, y2) = e−r(t)√
(y1)2+(y2)2, (36)
where
cos t=y1√
(y1)2 + (y2)2, sin t=
y2√(y1)2 + (y2)2
, tan t=y2
y1,
i.e. the level set {ϕ(y1, y2) ≡ 1} of the 1-homogeneous function ϕ in R2 is given by the theparametrized curve t→ (er(t) cos t, er(t) sin t).
Since the curvature κ of a smooth curve t→ (er(t) cos t, er(t) sin t) in R2 is
κ = − er√r2 + 1
(r − r2 − 1), (37)
the vanishing of the expression r−r2−1 means the infinitesimal linearity of the correspondingpositively homogeneous function in R2.
29
Definition 49 Let ϕ(y1, y2) be a positively 1-homogeneous function on R2 and let κ(t) bethe curvature of the curve t → (er(t) cos t, er(t) sin t) defined by the equations (36). We saythat ϕ(y1, y2) is strongly convex, if κ(t) 6= 0 for all t ∈ R.
Conditions (A), (B), (C) in the following theorem imply that the projective factor P atx0 ∈ M is a non-linear function, and hence, according to Remark 3, (M,F) is a non-Riemannian Finsler manifold.
Theorem 50 Let (M,F) be a projectively flat Finsler surface of non-zero constant curva-ture covered by a coordinate system (x1, x2). Assume that there exists a point x0 ∈M suchthat one of the following conditions hold
(A) F induces a scalar product on Tx0M and the projective factor P at x0 is a strongly
convex positively 1-homogeneous function,
(B) F(x0, y) is a strongly convex absolutely 1-homogeneous function on Tx0M , and the
projective factor P(x0, y) on Tx0M satisfies P(x0, y) = c·F(x0, y) with 0 6= c ∈ R,
(C) there is a projectively related Euclidean coordinate system of (M,F) centered at x0 andone has
F(0, y) = |y| ± 〈a, y〉 and P(0, y) =1
2(±|y| − 〈a, y〉) . (38)
Assume that the vector fields U = U i ∂∂xi , V = V i ∂
∂xi ∈ X∞(M) have constant coordinatefunctions and let ξ = R(U, V ) be the corresponding curvature vector field. Then the in-finitesimal holonomy algebra hol∗x(M) has an infinite dimensional subalgebra generated bythe vector fields ξ
∣∣x0
, ∇1ξ|x0 , ∇2ξ|x0 and ∇1 (∇2ξ) |x0 .
Proof. Since (M,F) is of constant flag curvature, we can write
Rijk(x, y) = λ(δijgkm(x, y)ym − δikgjm(x, y)ym
), with λ = const.
According to Lemma 2 the horizontal Berwald covariant derivative ∇WR of the tensor fieldR = Rijk(x, y)dxj ∧ dxk ∂
∂xi vanishes and hence ∇WR = 0.Since the curvature tensor field is skew-symmetric, R(x,y) acts on the one-dimensional wedgeproduct TxM ∧ TxM . The covariant derivative ∇W ξ of the curvature vector field ξ =R(U, V )= 1
2R(U ⊗ V −V ⊗ U)=R(U∧V ) can be written in the form
∇W ξ = R (∇W (U ∧ V )) = R(∇WU ∧ V + U ∧∇WV ).
We have U ∧ V = 12
(U1V 2 − U2V 1
)∂∂x1 ∧ ∂
∂x2 and hence
∇W ξ = (U1V 2 − V 1U2)W kR(∇k(∂∂x1 ∧ ∂
∂x2
) ), (39)
where ∇kξ := ∇ ∂
∂xkξ. Since
∇k(∂∂x1 ∧ ∂
∂x2
)=(∇k ∂
∂x1
)∧ ∂∂x2 + ∂
∂x1 ∧(∇k ∂
∂x2
)=
= Glk1∂∂xl ∧ ∂
∂x2 + ∂∂x1 ∧Gmk2
∂∂xm =
(G1k1 +G2
k2
)∂∂x1 ∧ ∂
∂x2
we obtain∇W ξ =
(G1k1 +G2
k2
)W kR(U, V ) =
(G1k1 +G2
k2
)W kξ.
Since the geodesic coefficients are given by (17) we have
∇W ξ = GmkmWkξ = 3
∂P∂yk
W kξ. (40)
30
Hence
∇Z (∇W ξ) = 3∇Z(∂P∂yk
W kξ
)= 3
{∇Z
(∂P∂yk
W k
)ξ +
(∂P∂yk
W k
)(∂P∂yl
Zl)}
ξ.
Let W be a vector field with constant coordinate functions. Then, using (17) we get
∇Z(∂P∂yk
W k
)=
(∂2P
∂xj∂yk−Gmj
∂2P∂ym∂yk
)W kZj =
(∂2P
∂xj∂yk− P ∂2P
∂yk∂yj
)W kZj ,
and hence
∇Z (∇W ξ) = 3
{∂2P
∂xj∂yk− P ∂2P
∂yk∂yj+∂P∂yk
∂P∂yj
}W kZjξ. (41)
Let x0 ∈ M be the point with coordinates (0, 0) in the local coordinate system of (M,F)corresponding to the canonical coordinates of the projectively related Euclidean plane. Ac-cording to Lemma 8.2.1 in [8], p.155, we have
∂2P∂x1∂y2
− P ∂2P∂y1∂y2
+∂P∂y1
∂P∂y2
= 2∂P∂y1
∂P∂y2− l
2
∂2F2
∂y1∂y2= 2
∂P∂y1
∂P∂y2− λ g12.
Hence the vector fields ξ∣∣x0
, ∇1ξ|x0 , ∇2ξ|x0 and ∇1 (∇2ξ) |x0 are linearly independent if andonly if the functions
1,∂P∂y1
∣∣∣x0
,∂P∂y2
∣∣∣x0
,
(2∂P∂y1
∂P∂y2− λ g12
) ∣∣∣x0
(42)
are linearly independent, where g12 = gy( ∂∂x1 ,
∂∂x2 ) is the component of the metric tensor of
(M,F).
Lemma 51 The functions ∂P(0,y)∂y1 , ∂P(0,y)
∂y2 and P(0, y)∂2P(0,y)∂y1∂y2 can be expressed in the polar
coordinate system (er, t) by
∂P(0, y)
∂y1=(cos t+ r sin t)e−r,
∂P(0, y)
∂y2=(sin t− r cos t)e−r,
P(0, y)∂2P(0, y)
∂y1∂y2=(r2 + 1− r)e−2rsin t cos t,
where the dot refers to differentiation with respect to the variable t.
Proof. We obtain from ∂e−r
∂y1 = −e−r r ∂t∂y1 and from − y2
(y1)2 = ∂∂y1 (y
2
y1 )d tan tdt
∂t∂y1 =
1cos2 t
∂t∂y1 that ∂e−r
∂y1 = e−r r cos2 t y2
(y1)2 = e−r r y2
(y1)2+(y2)2 . Hence
∂P(0, y)
∂y1=∂(e−r√
(y1)2 + (y2)2)
∂y1= e−r
(r
y2√(y1)2 + (y2)2
+y1√
(y1)2 + (y2)2
).
Similarly, we have ∂e−r
∂y2 = −e−r r cos2 t 1y1 = −e−r r y1
(y1)2+(y2)2 . Hence
∂P(0, y)
∂y2=∂(e−r√
(y1)2 + (y2)2)
∂y2= e−r
(− r y1√
(y1)2 + (y2)2+
y2√(y1)2 + (y2)2
).
Finally we have
∂2P(0, y)
∂y1∂y2=∂(sin t− r cos t)e−r
∂y1= (r − r2 − 1)e−r sin t cos t
1√(y1)2 + (y2)2
.
Replacing ϕ by the function P(0, y) in the expression (36) we get the assertion.
31
Lemma 52 Let r : R→ R be a 2π-periodic smooth function such that the inequality r(t)−r2(t)− 1 6= 0 holds on a dense subset of R. Then the functions
1, (cos t+ r sin t)e−r, (sin t− r cos t)e−r, (cos t+ r sin t)(sin t− r cos t)e−2r (43)
are linearly independent.
Proof. The derivative of (cos t+ r sin t)e−r and of (sin t− r cos t)e−r are (r− r2−1)e−r sin tand (r− r2−1)e−r cos t, respectively, hence the functions (43) do not vanish identically. Letus consider a linear combination
A+B(cos t+ r sin t)e−r + C(sin t− r cos t)e−r +D(cos t+ r sin t)(sin t− r cos t)e−2r = 0
with constant coefficients A,B,C,D. We differentiate and divide by e−t(r− r2 − 1) and wehave
B sin t− C cos t−D(cos 2t+ r sin 2t)e−r = 0.
Putting t = 0 and t = π we get C = −De−r(0) = De−r(π). Since e−r(0), e−r(π) > 0 we getC = D = 0 and hence A = B = C = D = 0.
Now, assume that condition (A) of Theorem 50 is fulfilled. According to Proposition 48 ifthe functions (42) are linearly independent, then the holonomy group Holx0
(M) is an infinitedimensional subgroup of Diff∞(Ix0
M). The function F(x0, y) induces a scalar product onTx0
M , consequently the component g12 of the metric tensor is constant on Tx0M . Hence
Holx0(M) is infinite dimensional if the functions
1,∂P∂y1
∣∣∣x0
,∂P∂y2
∣∣∣x0
,∂P∂y1
∂P∂y2
∣∣∣x0
(44)
are linearly independent. This follows from Lemma 52 and hence the assertion of the theoremis true.
Assume that condition (B) is satisfied. We denote ϕ(y) = F(x0, y). Using the expressions(42) we obtain that the vector fields ξ
∣∣x0
, ∇1ξ|x0 , ∇2ξ|x0 and ∇1 (∇2ξ) |x0 are linearlyindependent if and only if the functions
1,∂P∂y1
∣∣∣x0
= c∂ϕ
∂y1,
∂P∂y2
∣∣∣x0
= c∂ϕ
∂y2(2∂P∂y1
∂P∂y2− λ g12
) ∣∣∣x0
= (2c2 − λ)∂ϕ
∂y1
∂ϕ
∂y2− λ ϕ ∂2ϕ
∂y1∂y2
are linearly independent. According to Lemma 51 this is equivalent to the linear indepen-dence of the functions
1, (cos t+ r sin t) e−r, (sin t− r cos t) e−r,
(2c2 − λ)(cos t+ r sin t)(sin t− r cos t) e−2r − λ(r − r2 − 1) e−2r sin t cos t.
If r = const then these functions are 1, cos t e−r, sin t e−r, 2c2 cos t sin t e−2r, hence theassertion follows from Lemma 52. In the following we can assume that r(t) 6= const. Lett0 ∈ R such that r(t0) = 0 and κ(t0) 6= 0. We rotate the coordinate system at the angle −t0with respect to the euclidean norm
√(y1)2 + (y2)2, then we get in the new polar coordinate
system that r(0) = 0 and κ(0) 6= 0. Consider the linear combination
A+B(cos t+ r sin t)e−r + C(sin t− r cos t) e−r+
+D((2c2−λ)(cos t+r sin t)(sin t− r cos t) e−2r −λ(r−r2−1) e−2r sin t cos t
)=0
(45)
with some constants A,B,C,D. Since the function ϕ is absolutely homogeneous, the func-tion r(t) is π-periodic. Putting t+ π into t, the value of
A+D(2c2 − λ)(cos t+ r sin t)(sin t− r cos t) e−2r − λ(r − r2 − 1) e−2r sin t cos t
32
does not change, but the value of
B(cos t+ r sin t) e−r + C(sin t− r cos t) e−r
changes sign. Since Lemma 52 implies that (cos t + r sin t) e−r and (sin t − r cos t) e−r arelinearly independent, we have B = C = 0 and (45) becomes
Ae2r +D(
(2c2 − λ)[− r cos 2t+
1
2(1− r2) sin 2t
]− λ
2(r − r2 − 1) sin 2t
)= 0. (46)
Since r(0) = 0 at t = 0, we have A = 0. If D 6= 0 then (46) gives
(2c2 − λ)[− r cos 2t+
1
2(1− r2) sin 2t
]− λ
2(r − r2 − 1) sin 2t = 0.
By derivation and putting t = 0 we obtain
(2c2 − λ)[− r(0) + 1
]− λ(r(0)− 1) = 2c2(1− r(0)) = 0.
Using the relation (37) condition (B) gives κ(0) = er(0)(1− r(0)) 6= 0, which is a contradic-tion. Hence D = 0 and the vector fields ξ
∣∣x0
, ∇1ξ|x0 , ∇2ξ|x0 and ∇1(∇2ξ) |x0 are linearlyindependent. Using Proposition 48 we obtain the assertion.
Suppose now that the condition (C) holds. Hence we have
∂F∂y1
(0, y) =y1
|y|± a1,
∂F∂y2
(0, y) =y2
|y|± a2,
∂2F∂y1∂y2
(0, y) = −y1y2
|y|3,
and
g12 =
(y1
|y|± a1
)(y2
|y|± a2
)−(
1±⟨a,
y
|y|
⟩) y1y2
|y|2. (47)
Similarly, we obtain from condition (C) that
∂P∂y1
(0, y) = ± y1
|y|− a1,
∂P∂y2
(0, y) = ± y2
|y|− a2.
Using the expressions (42) we get that the vector fields ξ∣∣x0
, ∇1ξ|x0, ∇2ξ|x0
, ∇1 (∇2ξ) |x0
are linearly independent if and only if the functions
1,∂P∂y1
∣∣∣(0,y)
= ± y1
|y|− a1,
∂P∂y2
∣∣∣(0,y)
= ± y2
|y|− a2
and
2∂P∂y1
∂P∂y2−λ g12
∣∣∣(0,y)
= ∓⟨a,
y
|y|
⟩y1y2
|y|2+(1−λ)
y1y2
|y|2∓ (2+λ)
(a2y1
|y|+a1
y2
|y|
)+(2−λ)a1a2
are linearly independent. Putting
cos t =y1
|y|, sin t =
y2
|y|
we obtain that this condition is true, since the trigonometric polynomials
1, cos t, sin t, (1− λ) cos t sin t∓ (a1 cos t+ a2 sin t) cos t sin t
are linearly independent. Hence Holx0(M) is infinite dimensional.
33
6.2 Projective Finsler manifolds of constant curvature
Now we will prove that the infinitesimal holonomy algebra of a totally geodesic submanifoldof a Finsler manifold can be embedded into the infinitesimal holonomy algebra of the entiremanifold. This result yields a lower estimate for the dimension of the holonomy group.
6.2.1 Totally geodesic and auto-parallel submanifolds
Lemma 53 Let M be a totally geodesic submanifold in a spray manifold (M,S). Thecurvature vector fields at any point of M can be extended to a curvature vector field of M .
Proof. Assume that the manifolds M and M are k, respectively n = k + p dimen-sional. Let (x1, . . . , xk, xk+1, . . . , xn) be an adapted coordinate system, i. e. the sub-manifold M is locally given by the equations xk+1 = · · · = xn = 0. Using the notationof the proof of Lemma 1 we get from equation (12) that Gσα = 0 and Gσα β = 0 for any
(x1, . . . , xk, 0, . . . , 0; y1, . . . , yk, 0, . . . , 0) we have
∂Gσα∂xβ
−∂Gσβ∂xα
+GταGσβτ −GτβGσατ +GγαG
σβγ −G
γβG
σαγ = 0
at (x1, . . . , xk, 0, . . . , 0; y1, . . . , yk, 0, . . . , 0). Hence the curvature tensors K and K, corre-sponding to the spray S, respectively to the spray S satisfy
K(X,Y )(x, y) = K(X,Y )(x, y) if x ∈ M and y,X, Y ∈ TxM.
It follows that for any given X,Y ∈ TxM the curvature vector field ξ(y) = K(X,Y )(x, y) atx ∈ M defined on TxM can be extended to the curvature vector field ξ(y) = K(X,Y )(x, y)at x ∈ M defined on TxM .
Theorem 54 Let M be a totally geodesic 2-dimensional submanifold of a Finsler manifold(M,F) such that the infinitesimal holonomy algebra hol∗x(M) of M is infinite dimensional.Then the infinitesimal holonomy algebra hol∗x(M) of M is infinite dimensional.
Proof. According to Lemma 1 any curvature vector field of M at x ∈ M ⊂M defined onIxM can be extended to a curvature vector field on the indicatrix IxM . Hence the curvaturealgebra Rx(M) of the submanifold M can be embedded into the curvature algebra Rx(M)of the manifold (M,F). Assume that ξ is a vector field belonging to the infinitesimalholonomy algebra hol∗x(M) which can be extended to the vector field ξ belonging to theinfinitesimal holonomy algebra hol∗x(M). Any a vector field X ∈X∞(M) can be extendedto a vector field X ∈X∞(M), hence the horizontal Berwald covariant derivative along X ∈X∞(M) of ξ can be extended to the Berwald horizontal covariant derivative along X ∈X∞(M) of the vector field ξ. It follows that the infinitesimal holonomy algebra hol∗x(M) ofthe submanifold M can be embedded into the infinitesimal holonomy algebra hol∗x(M) ofthe Finsler manifold (M,F). Consequently, hol∗x(M) is infinite dimensional and hence theholonomy group Holx(M) is an infinite dimensional subgroup of Diff∞(IxM).
This result can be applied to locally projectively flat Finsler manifolds, as they have foreach tangent 2-plane a totally geodesic submanifold which is tangent to this 2-plane.
Corollary 55 If a locally projectively flat Finsler manifold has a 2-dimensional totallygeodesic submanifold satisfying one of the conditions of Theorem 50, then its infinitesimalholonomy algebra is infinite dimensional.
According to equations (18) and (19) the projectively flat Randers manifolds of non-zeroconstant curvature satisfy condition (C) of Theorem 50. We can apply Corollary 55 to thesemanifolds and we get the following
34
Theorem 56 The infinitesimal holonomy algebra of any projectively flat Randers manifoldsof non-zero constant flag curvature is infinite dimensional.
R. Bryant in [Br1], [Br2] introduced and studied complete Finsler metrics of positive cur-vature on S2. He proved that there exists exactly a 2-parameter family of Finsler metricson S2 with curvature = 1 with great circles as geodesics. Z. Shen generalized a 1-parameterfamily of complete Bryant metrics to Sn satisfying
F(0, y) = |y| cosα, P(0, y) = |y| sinα (48)
with |α| < π2 in a coordinate neighbourhood centered at 0 ∈ Rn, (cf. Example 7.1. in [37]
and Example 8.2.9 in [8]).We investigate the holonomy groups of two families of metrics, containing the 1-parameterfamily of complete Bryant-Shen metrics (48). The first family in the following theorem isdefined by condition (A), which is motivated by Theorem 8.2.3 in [8]. There is given thefollowing construction:If ψ = ψ(y) is an arbitrary Minkowski norm on Rn and ϕ = ϕ(y) is an arbitrary positively1-homogeneous function on Rn, then there exists a projectively flat Finsler metric F ofconstant flag curvature −1, defined on a neighbourhood of the origin, such that F and itsprojective factor P satisfy F(0, y) = ψ(y) and P(0, y) = ϕ(y).Condition (B) in the next theorem is confirmed by Example 7 in [37], p. 1726, where it isproved that for an arbitrary given Minkowski norm ϕ and |ϑ| < π
2 there exists a projectivelyflat Finsler function F of constant curvature = 1 defined on a neighbourhood of 0 ∈ Rn,such that
F(0, y) = ϕ(y) cosϑ and P(0, y) = ϕ(y) sinϑ.
Conditions (A) and (B) in Theorem 50 together with Corollary 55 yield the following
Theorem 57 Let (M,F) be a projectively flat Finsler manifold of non-zero constant cur-vature. Assume that there exists a point x0 ∈ M and a 2-dimensional totally geodesicsubmanifold M through x0 such that one of the following conditions holds
(A) F induces a scalar product on Tx0M , and the projective factor P on Tx0
M is a stronglyconvex positively 1-homogeneous function,
(B) F(x0, y) on Tx0M is a strongly convex absolutely 1-homogeneous function on Tx0
M ,and the projective factor P(x0, y) on Tx0
M satisfies P(x0, y) = c ·F(x0, y) with 0 6=c ∈ R.
Then the infinitesimal holonomy hol∗x(M) of M is infinite dimensional.
7 Dimension of the holonomy group
Let (M,F ) be a positive definite Finsler manifold and x ∈ M an arbitrary point in M .According to Proposition 3 of [25], the infinitesimal holonomy algebra hol∗x(M) is tangentto the holonomy group Holx(M). Therefore the group generated by the exponential im-age of the infinitesimal holonomy algebra at x ∈ M with respect to the exponential mapexpx :X∞(IxM) → Diff∞(IxM) is a subgroup of the closed holonomy group Holx(M) (seeTheorem 3.1 of [28]). Consequently, we have the following estimation on the dimensions:
dim hol∗x(M) ≤ dim Holx(M). (49)
Proposition 58 The infinitesimal holonomy algebra hol∗x(M) of any locally projectively flatnon-Riemannian Finsler surface (M,F) of constant curvature λ 6= 0 is infinite dimensional.
Proof. We use for the proof Lemma 48 and the notations introduced in the proof of Theorem50 in the previous section on projective Finsler surfaces of constant curvature. Using theassertion on the vector fields (42) we obtain
35
Lemma 59 For any fixed 1 ≤ j, k ≤ 2
y → ξ(x, y), y → ∇1ξ(x, y), y → ∇2ξ(x, y), y → ∇j(∇kξ) (x, y), (50)
considered as vector fields on IxM , are R-linearly independent if and only if the
1,∂P∂y1
,∂P∂y2
,∂2P
∂yj∂yk− λ
4gjk (51)
are linearly independent functions on TxM .
Since we assumed that the Finsler function F is non-Riemannian at the point x, i.e. F2(x, y)is non-quadratic in y, the function P(x, y) is non-linear in y on TxM (cf. eq. (17)). Let uschoose a direction y0 = (y1
0 , y20) ∈ TxM with y1
0 6= 0, y20 6= 0 and having property that P is
non-linear 1-homogeneous function in a conic neighbourhood U of y0 in TxM . By restrictingU if it is necessary we can suppose that for any y ∈ U we have y1 6= 0, y2 6= 0.To avoid confusion between coordinate indexes and exponents, we rename the fiber coor-dinates of vectors belonging to U by (u, v) = (y1, y2). Using the values of P on U we candefine a 1-variable function f = f(t) on an interval I ⊂ R by
f(t) :=1
vP(x1, x2, tv, v). (52)
Then we can express P and its derivatives with f :
P=v f(u/v),∂P∂y1
=f ′(u/v),∂P∂y2
=f(u/v)− u
vf ′(u/v),
∂2P∂y1∂y1
=1
vf ′′(u/v),
∂2P∂y1∂y2
=− u
v2f ′′(u/v),
∂2P∂y2∂y2
=u2
v3f ′′(u/v).
(53)
Lemma 60 The functions 1, ∂P∂y1 ,∂P∂y2 are linearly independent.
Proof. A nontrivial relation a+ b ∂P∂y1 + c ∂P∂y2 = 0 yields the differential equation a+ bf ′ +
c(f − tf ′) = 0. It is clear that both b and c cannot be zero. If c 6= 0 we get the differentialequation
(a+ cf)′
a+ cf=
1
t− bc
.
The solutions is f(t) = t− (a+b)/c and therefore the corresponding P(u, v) = u−v(a+b)/cis linear which is a contradiction. If c = 0, then b 6= 0 and f = −ab t+K. The correspondingP(u, v) = −abu+Kv is again linear which is a contradiction.
Let us assume now, that the infinitesimal holonomy algebra is finite dimensional. We willshow that this assumption leads to contradiction which will prove then, that the infinitesimalholonomy algebra is actually infinite dimensional.
Since IxM is 1-dimensional, according to the Lemma 48, the 4 vector fields in (50) arelinearly dependent for any j, k ∈ {1, 2}. Using Lemma 59 we get that the functions
1, P1, P2, PjPk −λ
4gjk (54)
(Pi = ∂P∂yi , Pjk = ∂2P
∂yj∂yk) are linearly dependent for any j, k ∈ {1, 2}. From Lemma 60
we know, that the first three functions in (54) are linearly independent. Therefore by theassumption, the fourth function must be a linear combination of the first three, that is thereexist constants ai, bi, ci ∈ R, i = 1, 2, 3, such that
λ
4g11 = P1P1 + a1 + b1P1 + c1P2,
λ
4g12 = P1P2 + a2 + b2P1 + c2P2,
λ
4g22 = P2P2 + a3 + b3P1 + c3P2.
(55)
36
Using (??) we get ∂1g21 − ∂2g11 = 0 and ∂1g22 − ∂2g12 = 0 which yield
P2P11 − P1P12 + b2P11 + (c2 − b1)P12 − c1P22 = 0,
P1P22 − P2P12 − b3P11 + (b2 − c3)P12 + c2P22 = 0.(56)
Using the expressions (53) we obtain from (56) the equations
(f − u
vf ′)
1
vf ′′ + f ′
u
v2f ′′ + b2
1
vf ′′ − (c2 − b1)
u
v2f ′′ − c1
u2
v3f ′′ = 0,
f ′u2
v3f ′′ + (f − u
vf ′)
u
v2f ′′ − b3
1
vf ′′ − (b2 − c3)
u
v2f ′′ + c2
u2
v3f ′′ = 0.
(57)
Since by the non-linearity of P on U we have f ′′ 6= 0, equations (57) can divide by f ′′/vand we get
f + b2 + (b1 − c2)u
v− c1u
2
v2= 0
u
vf − b3 + (c3 − b2)
u
v+
c2u2
v2= 0.
(58)
for any t = u/v in an interval I ⊂ R. The solution of this system of quadratic equationsfor the function f is f(t) = −c2 t − b2 with c1 = b3 = 0, b1 = 2c2, c3 = 2b2. But this isa contradiction, since we supposed that by the non-linearity of P we have f ′′ 6= 0 on thisinterval. Hence the functions 1, P1, P2, PjPk − λ
4 gjk can not be linearly dependent for anyj, k ∈ {1, 2}, from which follows the assertion.
Remark 61 From Proposition 58 we get that if (M,F) is non-Riemannian and λ 6= 0, thenthe holonomy group has an infinite dimensional tangent algebra.
Indeed, according to Theorem 6.3 in [25] the infinitesimal holonomy algebra hol∗x(M) istangent to the holonomy group Holx(M), from which follows the assertion.
Now, we can prove our main result:
Theorem 62 The holonomy group of a locally projectively flat simply connected Finslermanifold (M,F) of constant curvature λ is finite dimensional if and only if (M,F) is Rie-mannian or λ = 0.
Proof. If (M,F) is Riemannian then its holonomy group is a Lie subgroup of the orthog-onal group and therefore it is a finite dimensional compact Lie group. If (M,F) has zerocurvature, then the horizontal distribution associated to the canonical connection in thetangent bundle is integrable and hence the holonomy group is trivial.If (M,F) is non-Riemannian having non-zero curvature λ, then for each tangent 2-plane
S⊂TxM the manifold M has a totally geodesic submanifold M ⊂M such that TxM = S.This M with the induced metric is a locally projectively flat Finsler surface of constantcurvature λ. Therefore from Proposition 58 we get that hol∗x(M) is infinite dimensional.Moreover, according to Theorem 4.3 in [27], if a Finsler manifold (M,F) has a totally
geodesic 2-dimensional submanifold M such that the infinitesimal holonomy algebra of Mis infinite dimensional, then the infinitesimal holonomy algebra hol∗x(M) of the containingmanifold is also infinite dimensional. Using (49) we get that Holx(M) cannot be finitedimensional. Hence the assertion is true.
We note that there are examples of non-Riemannian type locally projectively flat Finslermanifolds with λ = 0 curvature, (cf. [22]).
Remark 63 In the discussion before the previous theorem, the key condition for the Finslermetric tensor was not the positive definiteness but its non-degenerate property. ThereforeTheorem 62 can be generalized as follows.
37
A pair (M,F) is called semi-Finsler manifold if in the definition of Finsler manifolds thepositive definiteness of the Finsler metric tensor is replaced by the nondegenerate property.Then we have
Corollary 64 The holonomy group of a locally projectively flat simply connected semi-Finsler manifold (M,F) of constant curvature λ is finite dimensional if and only if (M,F)is semi-Riemannian or λ = 0.
8 Maximal holonomy
8.1 Holonomy group as a subgroup of the diffeomorphism group ofthe indicatrix
The group Diff∞(K) of diffeomorphisms of a compact manifold K is an infinite dimensionalLie group belonging to the class of Frechet Lie groups. The Lie algebra of Diff∞(K) is theLie algebra X∞(K) of smooth vector fields on K endowed with the negative of the usual Liebracket of vector fields. Diff∞(K) is modeled on the locally convex topological Frechet vec-tor space X∞(K). A sequence {fj}j∈N ⊂ X∞(K) converges to f in the topology of X∞(K) ifand only if the vector fields fj and all their derivatives converge uniformly to f , respectivelyto the corresponding derivatives of f . We note that the difficulty of the theory of Frechetmanifolds comes from the fact that the inverse function theorem and the existence theoremsfor differential equations, which are well known for Banach manifolds, are not true in thiscategory. These problems have led to the concept of regular Frechet Lie groups (cf. H. Omori[32] Chapter III, A. Kriegl – P. W. Michor [21] Chapter VIII). The distinguishing propertiesof regular Frechet Lie groups can be summarized as the existence of smooth exponentialmap from the Lie algebra of the Frechet Lie groups to the group itself, and the existenceof product integrals, which produces the convergence of some approximation methods forsolving differential equations (cf. Section III.5. in [32], pp. 83 –89). In particular Diff∞(K)is a topological group which is an inverse limit of Lie groups modeled on Banach spaces andhence it is a regular Frechet Lie group (Corollary 5.4 in [32]).
Let H be a subgroup of the diffeomorphism group Diff∞(K) of a differentiable manifoldK. A vector field X ∈ X∞(K) is called tangent to H ⊂ Diff∞(K) if there exists a C1-differentiable 1-parameter family {Φ(t) ∈ H}t∈R of diffeomorphisms ofK such that Φ(0) = Id
and dΦ(t)dt
∣∣t=0
= X. A Lie subalgebra h of X∞(K) is called tangent to H, if all elements ofh are tangent vector fields to H.
We denote by (IM,π,M) the indicatrix bundle of the Finsler manifold (M,F), the indicatrixIxM at x ∈ M is the compact hypersurface IxM := {y ∈ TxM ; F(y) = 1} in TxM whichis diffeomorphic to the sphere Sn−1, if dim(M) = n. The homogeneous (nonlinear) paralleltranslation τc : Tc(0)M → Tc(1)M along a curve c : [0, 1] → M preserves the value of theFinsler function, hence it induces a map τc : Ic(0)M −→ Ic(1)M between the indicatrices.
The holonomy group Holx(M) of the Finsler manifold (M,F) at a point x ∈ M is the sub-group of the group of diffeomorphisms Diff∞(IxM) generated by homogeneous (nonlinear)parallel translations of IxM along piece-wise differentiable closed curves initiated at thepoint x ∈ M . The closed holonomy group is the topological closure Holx(M) of the holon-omy group with respect of the Frechet topology of Diff∞(IxM).We remark that the diffeomorphism group Diff∞(IxM) of the indicatrix IxM is a regularinfinite dimensional Lie group modeled on the vector space X∞(IxM). In this categorythe group structure is locally determined by the Lie algebra X∞(IxM) of the Lie groupDiff∞(IxM) (cf. [21, 32]).
For any vector fields X,Y ∈ X∞(M) on M the vector field ξ = R(X,Y ) ∈ X∞(IM) iscalled a curvature vector field of (M,F) (see [24]). The Lie algebra R(M) of vector fieldsgenerated by the curvature vector fields of (M,F) is called the curvature algebra of (M,F).
38
The restriction Rx(M) :={ξ∣∣IxM
; ξ ∈ R(M)}⊂ X∞(IxM) of the curvature algebra to an
indicatrix IxM is called the curvature algebra at the point x ∈M .
The infinitesimal holonomy algebra of (M,F) is the smallest Lie algebra hol∗(M) of vectorfields on the indicatrix bundle IM satisfying the following properties
a) any curvature vector field ξ belongs to hol∗(M),
b) if ξ ∈ hol∗(M) and X ∈ X∞(M) then the horizontal Berwald covariant derivative ∇Xξbelongs to hol∗(M).
The restriction hol∗x(M) :={ξ∣∣IxM
; ξ ∈ hol∗(M)}⊂ X∞(IxM) of the infinitesimal holon-
omy algebra to an indicatrix IxM is called the infinitesimal holonomy algebra at the pointx ∈M . One has R(M) ⊂ hol∗(M) and Rx(M) ⊂ hol∗x(M) for any x ∈M (see [25]).
Roughly speaking, the image of the curvature tensor (the curvature vector fields) deter-mines the curvature algebra, which generates (with the bracket operation and the covariantderivation) the infinitesimal holonomy algebra. Localising these object at a point x ∈M weobtain the curvature algebra and the infinitesimal holonomy algebra at x∈M .
The following assertion will be an important tool in the next discussion:
The infinitesimal holonomy algebra hol∗x(M) at any point x ∈M is tangent to the holonomygroup Holx(M). (Theorem 6.3 in [25]).
The holonomy group and its topological closure are interesting geometrical object whichreflects geometric properties of the Finsler manifold. In the characterization of the closedholonomy group we have the following corollary of Proposition 13:
Proposition 65 The group generated by the exponential image of the infinitesimal holon-omy algebra hol∗x(M) at a point x ∈M with respect to the exponential map exp : X∞(IxM)→Diff∞(IxM) is a subgroup of the closed holonomy group Holx(M).
8.2 The group Diff∞+ (S1) and the Fourier algebra
Let (M,F) be a Finsler 2-manifold. In this case the indicatrix is diffeomorphic to theunit circle S1, at any point x ∈ M . Moreover, if there exists a non-vanishing curvaturevector field at x ∈ M then any other curvature vector field at x ∈ M is proportional toit, which means that the curvature algebra is at most 1-dimensional. The infinitesimalholonomy algebra however, can be an infinite dimensional subalgebra of X∞(S1), thereforethe holonomy group can be an infinite dimensional subgroup of Diff∞+ (S1), cf. [27].
Let S1 = R mod 2π be the unit circle with the standard counterclockwise orientation.The group Diff∞+ (S1) of orientation preserving diffeomorphisms of the S1 is the connectedcomponent of Diff∞(S1). The Lie algebra of Diff∞+ (S1) is the Lie algebra X∞(S1) – denoted
also by Vect(S1) in the literature – can be written in the form f(t) ddt , where f is a 2π-periodic
smooth functions on the real line R. A sequence {fj ddt}j∈N ⊂ Vect (S1) converges to f ddt
in the Frechet topology of Vect(S1) if and only if the functions fj and all their derivativesconverge uniformly to f , respectively to the corresponding derivatives of f . The Lie bracketon Vect(S1) is given by [
fd
dt, gd
dt
]=(gdf
dt− dg
dtf) ddt.
The Fourier algebra F(S1) on S1 is the Lie subalgebra of Vect(S1) consisting of vector fieldsf ddt such that f(t) has finite Fourier series, i.e. f(t) is a Fourier polynomial. The vector
fields{ddt , cosnt ddt , sinnt ddt
}n∈N provide a basis for F(S1). A direct computation shows
that the vector fields
d
dt, cos t
d
dt, sin t
d
dt, cos 2t
d
dt, sin 2t
d
dt(59)
39
generate the Lie algebra F(S1). The complexification F(S1)⊗R C of F(S1) is called theWitt algebra W(S1) on S1 having the natural basis
{ieint ddt
}n∈Z, with the Lie bracket
[ieimt ddt , ieint d
dt ] = i(m− n)ei(n−m)t ddt .
Lemma 66 The group⟨
exp (F(S1))⟩
generated by the topological closure of the exponen-
tial image of the Fourier algebra F(S1) is the orientation preserving diffeomorphism groupDiff∞+ (S1).
Proof. The Fourier algebra F(S1) is a dense subalgebra of Vect(S1) with respect to theFrechet topology, i.e. F(S1) = Vect(S1). This assertion follows from the fact that any r-times continuously differentiable function can be approximated uniformly by the arithmeticalmeans of the partial sums of its Fourier series (cf. [16], 2.12 Theorem). The exponentialmapping is continuous (c.f. Lemma 4.1 in [32], p. 79), hence we have
exp(Vect(S1)
)= exp
(F(S1)
)⊂ exp
(F(S1)
)⊂ Diff∞+ (S1) (60)
which gives for the generated groups the relations⟨exp
(Vect(S1)
)⟩⊂⟨
exp(F(S1)
) ⟩⊂ Diff∞+ (S1). (61)
Moreover, the conjugation map Ad : Diff∞+ (S1)× Vect(S1) satisfies the relation
h exp sξ h−1 = exp sAd(h)ξ
for every h ∈ Diff∞+ (S1) and ξ ∈ Vect(S1). Clearly, the Lie algebra Vect(S1) is invariantunder conjugation and hence the group
⟨exp
(Vect(S1)
)⟩is also invariant under conjugation.
Therefore⟨exp
(Vect(S1)
)⟩is a non-trivial normal subgroup of Diff∞+ (S1). On the other hand
Diff∞+ (S1) is a simple group (cf. [15]) which means that its only non-trivial normal subgroupis itself. Therefore, we have
⟨exp
(Vect(S1)
)⟩= Diff∞+ (S1), and using (61) we get⟨
exp(F(S1))⟩
= Diff∞+ (S1).
8.3 Holonomy of the standard Funk plane and the Bryant-Shen2-spheres
Using the results of the preceding chapter we can prove the following statement, which pro-vides a useful tool for the investigation of the closed holonomy group of Finsler 2-manifolds.
Proposition 67 If the infinitesimal holonomy algebra hol∗x(M) at a point x ∈M of a simplyconnected Finsler 2-manifold (M,F) contains the Fourier algebra F(S1) on the indicatrix atx, then Holx(M) is isomorphic to Diff∞+ (S1).
Proof. Since M is simply connected we have
Holx(M) ⊂ Diff∞+ (S1). (62)
On the other hand, using Proposition 13, we get
exp(F(S1)
)⊂Holx(M) ⇒ exp
(F(S1)
)⊂Holx(M) ⇒
⟨exp
(F(S1)
) ⟩⊂Holx(M),
and from the last relation, using Lemma 66, we can obtain that
Diff∞+ (S1) ⊂ Holx(M). (63)
Comparing (62) and (63) we get the assertion.
Using this proposition we can prove our main result:
40
Theorem 68 Let (M,F) be a simply connected projectively flat Finsler manifold of constantcurvature λ 6= 0. Assume that there exists a point x0 ∈M such that on Tx0M the inducedMinkowski norm is an Euclidean norm, that is F(x0, y) = ‖y‖, and the projective factor atx0 satisfies P(x0, y) = c·‖y‖ with c ∈ R, c 6= 0. Then the closed holonomy group Holx0
(M)at x0 is isomorphic to Diff∞+ (S1).
Proof. Since (M,F) is a locally projectively flat Finsler manifold of non-zero constantcurvature, we can use an (x1, x2) local coordinate system centered at x0 ∈M , correspondingto the canonical coordinates of the Euclidean space which is projectively related to (M,F).Let (y1, y2) be the induced coordinate system in the tangent plane TxM . In the sequelwe identify the tangent plane Tx0
M with R2 by using the coordinate system (y1, y2). Wewill use the Euclidean norm ‖(y1, y2)‖ =
√(y1)2 + (y2)2 of R2 and the corresponding polar
coordinate system (er, t), too.
Let us consider the curvature vector field ξ at x0 = 0 defined by
ξ=R
(∂
∂x1,∂
∂x2
) ∣∣∣x=0
= λ(δi2g1m(0, y)ym − δi1g2m(0, y)ym
) ∂
∂xi
Since (M,F) is of constant flag curvature, the horizontal Berwald covariant derivative ∇WRof the tensor field R vanishes, c.f. Lemma 2. Therefore the covariant derivative of ξ can bewritten in the form
∇W ξ = R
(∇k(
∂
∂x1∧ ∂
∂x2
))W k.
Since
∇k(
∂
∂x1∧ ∂
∂x2
)=(G1k1 +G2
k2
) ∂
∂x1∧ ∂
∂x2
we obtain ∇W ξ =(G1k1 +G2
k2
)W kξ. Using (17) we can express Gmkm = 3 ∂P
∂yk= 3c y
k
‖y‖ and
hence
∇kξ = 3∂P
∂ykξ = 3c
yk
‖y‖ξ,
where we use the notation ∇k = ∇ ∂
∂xk. Moreover we have
∇j(∂P∂yk
)=
∂2P∂xj∂yk
−Gmj∂2P
∂ym∂yk=
∂2P∂xj∂yk
− P ∂2P∂yk∂yj
,
and hence
∇j (∇kξ) = 3
{∂2P
∂xj∂yk− P ∂2P
∂yk∂yj+ 3
∂P∂yk
∂P∂yj
}ξ.
According to Lemma 8.2.1, equation (8.25) in [8], p. 155, we obtain
∂2P∂xj∂yk
=∂P∂yj
∂P∂yk
+ P ∂2P∂yj∂yk
− λ
2
∂2F2
∂yj∂yk.
Using the assumptions on F and on the projective factor P we can get at x0
∇j (∇kξ) = 3
(4c2
∂F∂yj
∂F∂yk
− λ
2
∂2F2
∂yj∂yk
)ξ
and hence
∇j (∇kξ) = 3
(4 c2
yjyk
‖y‖2− λ δjk
)ξ,
where δjk ∈ {0, 1} such that δjk = 1 if and only if j = k.Let us introduce polar coordinates y1 = r cos t, y2 = r sin t in the tangent space Tx0M . We
41
can express the curvature vector field, its first and second covariant derivatives along theindicatrix curve {(cos t, sin t); 0 ≤ t < 2π} as follows:
ξ=λd
dt, ∇1ξ=3cλ cos t
d
dt, ∇2ξ=−3cλ sin t
d
dt, ∇1(∇2ξ) = 12 c2λ sin 2t
d
dt,
∇1(∇1ξ)=λ(12 c2 cos2 t−λ
) ddt, ∇2(∇2ξ)=λ
(12 c2 sin2 t−λ
) ddt.
Since c λ 6= 0, the vector fields
d
dt, cos t
d
dt, sin t
d
dt, cos t sin t
d
dt, cos2 t
d
dt, sin2 t
d
dt
are contained in the infinitesimal holonomy algebra hol∗x0(M). It follows that the generator
system {d
dt, cos t
d
dt, sin t
d
dt, cos 2t
d
dt, sin 2t
d
dt
}of the Fourier algebra F(S1) (c.f. equation (59)) is contained in the infinitesimal holonomyalgebra hol∗x0
(M). Hence the assertion follows from Proposition 67.
We remark, that the standard Funk plane and the Bryant-Shen 2-spheres are connected,projectively flat Finsler manifolds of nonzero constant curvature. Moreover, in each of them,there exists a point x0 ∈M and an adapted local coordinate system centered at x0 with thefollowing properties: the Finsler norm F(x0, y) and the projective factor P(x0, y) at x0 aregiven by F(x0, y) = ‖y‖ and by P(x0, y) = c·‖y‖ with some constant c ∈ R, c 6= 0, where‖y‖ is an Euclidean norm in the tangent space at x0. Using Theorem 68 we can obtain
Theorem 69 The closed holonomy groups of the standard Funk plane and of the Bryant-Shen 2-spheres are maximal, that is diffeomorphic to the orientation preserving diffeomor-phism group of S1.
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