Special connections on symplectic manifolds

Lorenz J. Schwachhofer1,2

April 26, 2003

Abstract

On a given symplectic manifold, there are many symplectic connections, i.e. torsion freeconnections w.r.t. which the symplectic form is parallel. We call such a connection specialif it is either the Levi-Civita connection of a Bochner-Kahler metric of arbitrary signature,a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with specialsymplectic holonomy.

We link these special connections to parabolic contact geometry, showing that the symplecticreduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is specialsymplectic in a canonical way. Moreover, we show that any special symplectic manifold ororbifold is locally equivalent to one of these symplectic reductions.

As a consequence, we are able to prove a number of rigidity results and other global prop-erties.

Keywords: Bochner-Kahler metrics, Ricci type connections, Symplectic holonomyMSC: 53D35, 53D05, 53D10

1 Introduction

Torsion free connections on a differentiable manifold M which preserve a given geometric structureare among the basic objects of interest in differential geometry. For example, if M carries aRiemannian metric, then there is a unique torsion free connection which is compatible with thismetric, called the Levi-Civita connection. Thus, every feature of the connection reflects a propertyof the metric structure.

In contrast, for a symplectic manifold (M,ω), there are many symplectic connections, where wecall a connection on M symplectic if it is torsion free and ω is parallel. Thus, in order to investigate’meaningful’ symplectic connections, we have to impose further conditions.

In [CS], the notion of a special symplectic connection was established. Special symplectic con-nections are defined as symplectic connections on a manifold of dimension at least 4 which belongto one of the following seemingly unrelated classes.

1Universitat Dortmund, Vogelpothsweg 87, 44221 Dortmund, Germany. e-mail: lschwach@math.uni-dortmund.de2Research supported by the Communaute francaise de Belgique, through an Action de Recherche Concertee de la

Direction de la Recherche Scientifique, and also by the Schwerpunktprogramm Globale Differentialgeometrie of theDeutsche Forschungsgesellschaft.

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1. Bochner-Kahler and Bochner-bi-Lagrangian connections

If the symplectic form is the Kahler form of a (pseudo-)Kahler metric, then its curvaturedecomposes into the Ricci curvature and the Bochner curvature ([Bo]). If the latter vanishes,then (the Levi-Civita connection of) this metric is called Bochner-Kahler.

Similarly, if the manifold is equipped with a bi-Lagrangian structure, i.e. two complementaryLagrangian distributions, then the curvature of a symplectic connection for which both dis-tributions are parallel decomposes into the Ricci curvature and the Bochner curvature. Sucha connection is called Bochner-bi-Lagrangian if its Bochner curvature vanishes.

For results on Bochner-Kahler and Bochner-bi-Lagrangian connections, see [Br2] and [K] andthe references cited therein. We shall give a brief summary of these structures in section 2.2.

2. Connections of Ricci type

Under the action of the symplectic group, the curvature of a symplectic connection decom-poses into two irreducible summands, namely the Ricci curvature and a Ricci flat component.If the latter component vanishes, then the connection is said to be of Ricci type.

Connections of Ricci type are critical points of a certain functional on the moduli space ofsymplectic connections ([BC1]). Furthermore, the canonical almost complex structure on thetwistor space induced by a symplectic connection is integrable iff the connection is of Riccitype ([BR], [V]). For further properties see also [CGR], [CGHR], [BC2], [CGS]. We shalltreat these connections in section 2.1.

3. Connections with special symplectic holonomy

A symplectic connection is said to have special symplectic holonomy if its holonomy is con-tained in a proper absolutely irreducible subgroup of the symplectic group.

The special symplectic holonomies have been classified in [MS] and further investigated in[Br1], [CMS], [S1], [S2], [S3]. These connections shall be discussed in section 2.3.

At first, it may seem unmotivated to collect all these structures in one definition, but we shallprovide ample justification for doing so. Indeed, our main results show that there is a beautifullink between special symplectic connections and parabolic contact geometry.

For this, consider a simple Lie group G with Lie algebra g. We say that g is 2-gradable, if g

contains the root space of a long root. In this case, the projectivization of the adjoint orbit ofa maximal root vector C ⊂ P

o(g) carries a canonical G-invariant contact structure. Here, Po(V )

denotes the set of oriented lines through 0 of a vector space V , so that Po(V ) is diffeomorphic

to a sphere. Each a ∈ g induces an action field a∗ on C with flow Ta := exp(Ra) ⊂ G, whichhence preserves the contact structure on C. Let Ca ⊂ C be the open subset on which a∗ is positivelytransversal to the contact distribution. We can cover Ca by open sets U such that the local quotientMU := Tloca \U , i.e. the quotient of U by a sufficiently small neighborhood of the identity in Ta, is amanifold. Then MU inherits a canonical symplectic structure. Our first main result is the following(cf. Theorem 3.11).

Theorem A: [CS] Let g be a simple 2-gradable Lie algebra with dim g ≥ 14, and let C ⊂ Po(g) be

the projectivization of the adjoint orbit of a maximal root vector. Let a ∈ g be such that Ca ⊂ C isnonempty, and let Ta = exp(Ra) ⊂ G. If for an open subset U ⊂ Ca the local quotient MU = Tloc

a \Uis a manifold, then MU carries a special symplectic connection.

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The dimension restriction on g guarantees that dimMU ≥ 4 and rules out the Lie algebras oftype A1, A2 and B2.

The type of special symplectic connection on MU is determined by the Lie algebra g. Infact, there is a one-to-one correspondence between the various conditions for special symplecticconnections and simple 2-gradable Lie algebras. More specifically, if the Lie algebra g is of typeAn, then the connections in Theorem A are Bochner-Kahler of signature (p, q) if g = su(p+1, q+1)or Bochner-bi-Lagrangian if g = sl(n,R); if g is of type Cn, then g = sp(n,R) and these connectionsare of Ricci type; if g is a 2-gradable Lie algebra of one of the remaining types, then the holonomy ofMU is contained in one of the special symplectic holonomy groups. Also, for two elements a, a′ ∈ g

for which Ca, Ca′ ⊂ C are nonempty, the corresponding connections from Theorem A are equivalentiff a′ is G-conjugate to a positive multiple of a.

Surprisingly, the connections from Theorem A exhaust all special symplectic connections, atleast locally. Namely we have the following

Theorem B: [CS] Let (M,ω) be a symplectic manifold with a special symplectic connection of classC4, and let g be the Lie algebra associated to the special symplectic condition as above.

1. Then there is a principal T-bundle M → M , where T is a one dimensional Lie group whichis not necessarily connected, and this bundle carries a principal connection with curvature ω.

2. Let T ⊂ T be the identity component. Then there is an a ∈ g such that T ∼= Ta ⊂ G, and aTa-equivariant local diffeomorphism ı : M → Ca which for each sufficiently small open subsetV ⊂ M induces a connection preserving diffeomorphism ı : Tloc\V → Tloc

a \U = MU , whereU := ı(V ) ⊂ Ca and MU carries the connection from Theorem A.

The situation in Theorem B can be illustrated by the following commutative diagram, wherethe vertical maps are quotients by the indicated Lie groups, and T\M →M is a regular covering.

M

T

ı //

T

Ca

Ta

M T\Mı //oo Ta\Ca

In fact, one might be tempted to summarize Theorems A and B by saying that for each a ∈ g, thequotient Ta\Ca carries a canonical special symplectic connection, and the map ı : T\M → Ta\Cais a connection preserving local diffeomorphism. If Ta\Ca is a manifold or an orbifold, then this isindeed correct. In general, however, Ta\Ca may be neither Hausdorff nor locally Euclidean, henceone has to formulate these results more carefully.

As consequences, we obtain the following

Corollary C: [CS] All special symplectic connections of C4-regularity are analytic, and the localmoduli space of these connections is finite dimensional, in the sense that the germ of the connec-tion at one point up to 3rd order determines the connection entirely. In fact, the generic specialsymplectic connection associated to the Lie algebra g depends on (rk(g) − 1) parameters.

Moreover, the Lie algebra s of vector fields on M whose flow preserves the connection is iso-morphic to stab(a)/(Ra) with a ∈ g from Theorem B, where stab(a) = x ∈ g | [x, a] = 0. Inparticular, dim s ≥ rk(g) − 1 with equality implying that s is abelian.

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When counting the parameters in the above corollary, we regard homothetic special symplecticconnections as equal, i.e. (M,ω,∇) is considered equivalent to (M,et0ω,∇) for all t0 ∈ R.

We can generalize Theorem B and Corollary C easily to orbifolds. Indeed, if M is an orbifoldwith a special symplectic connection, then we can write M = T\M where M is a manifold andT is a one dimensional Lie group acting properly and locally freely on M , and there is a localdiffeomorphism ı : M → Ca with the properties stated in Theorem B.

We also address the question of the existence of compact manifolds with special symplecticconnections. In the simply connected case, compactness already implies that the connection ishermitian symmetric. More specifically, we have the following

Theorem D: [CS] Let M be a compact simply connected manifold with a special symplectic con-nection of class C4. Then M is equivalent to one of the following hermitian symmetric spaces.

1. M ∼= (CPp × CP

q, ((q + 1)g0,−(p + 1)g0)), where g0 is the Fubini-Study metric. These areBochner-Kahler metrics of signature (p, q). Moreover, M ∼= (CP

n, g0) is also of Ricci type.

2. M ∼= SO(n + 2)/(SO(2) · SO(n)), whose holonomy is contained in the special symplecticholonomy group SL(2,R) · SO(n) ⊂ Aut(R2 ⊗ R

n).

3. M ∼= SU(2n + 2)/S(U(2) · U(2n)), whose holonomy is contained in the special symplecticholonomy group Sp(1) · SO(n,H) ⊂ Aut(Hn).

4. M ∼= SO(10)/U(5), whose holonomy is contained in the special symplectic holonomy groupSU(1, 5) ⊂ GL(20,R).

5. M ∼= E6/(U(1) · Spin(10)), whose holonomy is contained in the special symplectic holonomygroup Spin(2, 10) ⊂ GL(32,R).

In particular, there are no compact simply connected manifolds with any of the remainingtypes of special symplectic connections, i.e. M can be neither Bochner-bi-Lagrangian, nor can theholonomy of M be contained in any of the remaining special symplectic holonomies.

This paper is structured as follows. Following this introduction, we first describe the varioustypes of special symplectic connections and review some of their geometric features. We then showhow on a formal level, we can link these conditions to the algebraic formalism of 2-gradable simpleLie algebras. In section 3, we describe 2-gradable Lie algebras and perform the symplectic reduc-tion, showing that the local quotients Tloc

a \Ca carry special symplectic connections in a canonicalway, and hence prove Theorem A. In section 4, we investigate the structure equations of specialsymplectic connections and derive results which culminate in Theorem B. Finally, in section 5 weshow the existence of connection preserving vector fields and Corollary C, and the rigidity resultfrom Theorem D.

This report is closely related to the reference [CS]. In fact, the main results are shown in thatpaper, and we shall refer to it for many of the proofs. In this report, we emphasize the more concretedescription of special symplectic connections and their various geometrical features as opposed tothe more abstract construction from [CS], and we describe the link to parabolic contact geometryin more detail.

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2 Symplectic connections

Let (M,ω) be a symplectic manifold. A symplectic connection is a connection on the (tangentbundle of) M which is torsion free and for which ω is parallel.

It is not hard to see that symplectic connections exist on any symplectic manifold (M,ω).Namely, by Darboux’s theorem, around each point inM we can find a coordinate system (x1, . . . , xn)in which

ω = ωijdxi ∧ dxj ,

where (ω)i,j is a constant skew symmetric non-degenerate matrix. Then a connection is symplecticiff the Christoffel symbols Γkij have the property that the tensor

σijk :=∑

l

ωilΓljk

is totally symmetric. Thus, there are many symplectic connections. In coordinate free notation,this is seen by the observation that for two symplectic connections ∇ and ∇′ the tensor

σ(X,Y,Z) := ω(∇′XY −∇XY,Z) (1)

is totally symmetric, i.e. σ ∈ Γ(S3(TM)). Conversely, given a symplectic connection ∇ andσ ∈ Γ(S3(TM)), then (1) determines the symplectic connection ∇′. Thus, the space of symplecticconnections is an affine space whose linear part is given by the sections of S3(TM). In particular,this space is infinite dimensional, even if we take the quotient by the action of the symplectomor-phism group.

This may be one of the reasons why symplectic connections in their full generality are difficultto be utilized in order to obtain information on the underlying symplectic manifold. Thus, it isnatural to put certain ’reasonable’ restrictions on their curvature and thereby single out symplecticconnections with special behaviour.

In order to make this more precise, we recall the following terminology. For a given Lie subal-gebra h ⊂ End(V ) we define the space of formal curvature maps as

K(h) := R ∈ Λ2V ∗ ⊗ h | R(x, y)z +R(y, z)x +R(z, x)y = 0 for all x, y, z ∈ V .

This terminology is due to the fact that the curvature map of a torsion free connection alwayssatisfies the first Bianchi identity. Thus, Rp ∈ K(hp) where hp ⊂ End(TpM) is a subalgebra whichcontains the holonomy algebra of the connection. Note that K(h) is an h-module in an obviousway, with the action given by the formula

(h ·R)(x, y) := [h,R(x, y)] −R(hx, y) −R(x, hy) for all h ∈ h and x, y ∈ V .

There is an h-equivariant map Ric : K(h) → V ∗ ⊗ V ∗, given by

Ric(R)(x, y) := tr(R( , x)y).

The Bianchi identity implies that Ric(R)(x, y) − Ric(R)(y, x) = −trR(x, y), hence the image ofRic lies in S2(V ∗) ⊂ V ∗ ⊗ V ∗ if h ⊂ End(V ) consists of trace free endomorphisms.

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2.1 Connections of Ricci type

In general, given a symplectic vector space (V, ω), i.e. ω ∈ Λ2V ∗ is non-degenerate, we define thesymplectic group Sp(V, ω) and the symplectic Lie algebra sp(V, ω) by

Sp(V, ω) := g ∈ Aut(V ) | ω(gx, gy) = ω(x, y) for all x, y ∈ V ,

sp(V, ω) := h ∈ End(V ) | ω(hx, y) + ω(x, hy) = 0 for all x, y ∈ V .

Then Sp(V, ω) is a Lie group with Lie algebra sp(V, ω).If ∇ is a symplectic connection, then ωp ∈ Λ2TpM is invariant under the holonomy group, hence

Holp ⊂ Sp(TpM,ωp) so that at each point p ∈M the curvature Rp ∈ K(sp(TpM,ωp)).Note that sp(V, ω) consists of trace free endomorphisms, so that we have the sp(V, ω)-equivariant

mapRic : K(sp(V, ω)) −→ S2(V ∗).

More explicitly, this map is given by the following

Lemma 2.1 Let R ⊂ K(sp(V, ω)). Then Ric(R)(x, y) = ω(R(ω−1)x, y).

Proof. Let (ei, fi) be a basis of V such that, using the summation convention, ω−1 = ei∧fi. Thus,

Ric(R)(x, y) = tr(R( , x)y) = ω(R(ei, x)y, fi) − ω(R(fi, x)y, ei)

= ω(R(ei, x)fi, y) + ω(R(x, fi)ei, y) = ω(R(ei, fi)x, y).

Note that as a Sp(V, ω)-module, we have V ∼= V ∗ and sp(V, ω) ∼= S2V ∗ ∼= S2V , with anisomorphism for the latter being given by the equivariant map

: S2V −→ sp(V, ω), (x y)z := ω(x, z)y + ω(y, z)x for all x, y, z ∈ V . (2)

In fact, by virtue of Lemma 2.1 we may reinterpret the map Ric as

Ric : K(sp(V, ω)) −→ sp(V, ω), R 7−→ R(ω−1).

Now we make the following

Definition 2.2 A Lie subalgebra h ⊂ sp(V, ω) is called special symplectic if there is an h-equivariantlinear map : S2(V ) → h satisfying

(x y)z − (x z)y = 2 ω(y, z)x − ω(x, y)z + ω(x, z)y for all x, y, z ∈ V . (3)

By (2), it is evident that sp(V, ω) is special symplectic. Moreover, it is straightforward to verifythe following important fact.

Proposition 2.3 Let h ⊂ sp(V, ω) be a special symplectic subalgebra. Then for each ρ ∈ h, themap

Rρ : Λ2V −→ sp(V, ω), Rρ(x, y) := 2ω(x, y)ρ + x ρy − y ρx. (4)

satisfies the first Bianchi identity, i.e. Rρ ∈ K(h).

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From this, we now obtain the following

Proposition 2.4 For ρ ∈ sp(V, ω), we have Ric(Rρ)(x, y) = (dimV + 2)ω(ρx, y). Moreover, as aSp(V, ω)-module, we have the decomposition

K(sp(V, ω)) = R⊕W,

whereR := Rρ | ρ ∈ sp(V, ω) ∼= sp(V, ω), and W := ker(Ric).

Thus, R is irreducible. Moreover, dimW = 18(n−2)n(n+1)(n+3) where n := dimV . In particular,

W 6= 0 iff dimV ≥ 4, and in this case, W is irreducible as well.

Proof. Since sp(V, ω) is special, it follows that Rρ ∈ K(sp(V, ω)) by Proposition 2.3. Moreover,it is straightforward to verify that Rρ(ω

−1) = (n + 2)ρ which implies the second assertion. Inparticular, Ric : K(sp(V, ω)) → S2V ∗ is surjective, so that the asserted decomposition follows. Ris irreducible as sp(V, ω) is simple. Now consider the sequence of End(V )-equivariant linear maps

0 −→ S4V ∗ −→ S3V ∗ ⊗ V ∗ −→ S2V ∗ ⊗ Λ2V ∗ −→ V ∗ ⊗ Λ3V ∗ −→ Λ4V ∗ −→ 0,

which are given by skew symmetrization. Since we can regard these maps as the exterior differen-tiation of differential forms on V with polynomial coefficients, it follows easily that this sequenceis exact.

Let us now consider the map B : Λ2V ∗⊗sp(V, ω) → Λ3V ∗⊗V which is given by B(R)(x, y, z) :=R(x, y)z +R(y, z)x+R(z, x)y. Thus, K(sp(V, ω)) = ker(B).

Using the isomorphism sp(V, ω) ∼= S2V ∗ from (2), it is easy to verify that B : Λ2V ∗ ⊗ S2V ∗ →Λ3V ∗ ⊗ V ∗ coincides up to a multiple with the differential in the above exact sequence, henceK(sp(V, ω)) ∼= (S3V ∗⊗V ∗)/S4V ∗, and from this, the assertions follow from a dimension count and

by standard representation theoretical arguments.

By virtue of this proposition, we can now decompose

K(sp(TpM,ωp)) ∼= Rp ⊕Wp. (5)

for each p ∈M , and we make the following

Definition 2.5 Let (M,ω,∇) be a symplectic manifold with a symplectic connection. We say that∇ is of Ricci type if its curvature satisfies R∇

p ∈ Rp for all p ∈M with the decomposition (5).

If ∇ is of Ricci type, then there is a unique section ρ of endomorphisms of the tangent spacesof M for which the curvature of ∇ at each point has the form (4).

We shall now give a more geometric interpretation of connections of Ricci type. For this,consider a symplectic manifold (M,ω), and define its twistor space as

Z :=

Jp ∈ sp(TpM,ωp) | J2p = −Id, ωp(Jp , ) is positive definite.

The fibers of the canonical fibration π : Z →M can be identified with the hermitian symmetricspace Sp(n,R)/U(n) and hence carry a canonical complex structure. Also, if ∇ is a connection on

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M , then ∇ induces a connection on End(TM) = T ∗M ⊗ TM , and evidently, Z ⊂ End(TM) isparallel w.r.t. ∇. Thus, ∇ induces a decomposition of the tangent space

TJpZ = VJp ⊕HJp,

where VJp = ker(dπJp) is the vertical space. We now define the almost complex structure J on Zby the requirement that

1. J(HJp) = HJp and J(VJp) = VJp for all Jp ∈ Z,

2. J |VJpcoincides with the complex structure of π−1(p) ∼= Sp(n,R)/U(n),

3. dπ : (HJp , J) → (TpM,Jp) is complex linear.

Then the following is known.

Proposition 2.6 [BR] [V] Let ∇ be a symplectic conection on (M,ω). Then the almost complexstructure on the twistor space Z →M which is induced by ∇ is integrable iff ∇ is of Ricci type.

2.2 Bochner-Kahler and Bochner-bi-Lagrangian connections

Suppose that ∇ is the Levi-Civita connection of a Kahler metric g on M with Kahler form ω. Here,we use the term Kahler metric for a metric of arbitrary signature unless explicitly stated otherwise.Thus, there is a complex structure J on M with g(x, y) = ω(Jx, y) for all x, y ∈ TM , and J isparallel w.r.t. ∇. It follows that the curvature of ∇ takes values in the Lie algebra u(p, q), where(p, q) is the complex signature of the Kahler metric.

A bi-Lagrangian structure on a symplectic manifold (M,ω) is a splitting of the tangent bundleTM = L1 ⊕ L2 where Li are Lagrangian distributions. A symplectic connection ∇ on such anM is called bi-Lagrangian if both distributions are parallel. We can define a section J of theendomorphism bundle by J |Li

= (−1)iIdLi. Evidently, Jp ∈ sp(TpM,ωp) for all p ∈ M , and

J2 = Id. Conversely, given a section J with Jp ∈ sp(TpM,ωp) for all p ∈M and J2 = Id as above,we get a splitting of TM in to the Eigenspaces of J , and it follows that both of them must beisotropic and hence Lagrangian. Evidently, J is parallel w.r.t. any bi-Lagrangian connection ∇,hence the curvature of ∇ takes values in the Lie algebra u′(m) ⊂ sp(m,R) which is defined as thestabilizer of the endomorphism J ∈ sp(m,R) with J2 = Id.

Therefore, in order to determine the curvature spaces K(u(p, q)) and K(u′(m)) simultaneously,we observe that they are all of the form hJ := x ∈ sp(m,R) | [x, J ] = 0, where in the case ofu(p, q), J ∈ sp(m,R) is the invariant complex structure so that J2 = −Id, and in the case of u′(m),J ∈ sp(m,R) is the endomorphism with J2 = Id.

We define the circle product

: S2(V ) −→ hJ , (xy)z := ω(x, z)y+ω(y, z)x−ε(ω(Jx, z)Jy+ω(Jy, z)Jx+ω(Jx, y)Jz), (6)

where ε ∈ ±1 is given by J2 = εId. Indeed, it is straightforward to verify that ω((xy)z,w)+ω(z, (x y)w) = 0 and (x y)Jz = J(x y)z for all x, y, z, w ∈ V , so that x y ∈ hJ in either case.Moreover, also satisfies (3), so that hJ ⊂ sp(V, ω) is a special symplectic subalgebra in the senseof Definition 2.2, hence by Proposition 2.3 the map Rρ : Λ2V → hJ from (4) is an element of K(hJ )for all ρ ∈ hJ .

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Proposition 2.7 Let hJ ⊂ sp(V, ω) be one of the special symplectic subalgebras u(p, q) or u′(m)with the product : S2(V ) → hJ from (6), and let J ∈ sp(V, ω) be the endomorphism with J2 =ε IdV , ε = ±1, which is stabilized by hJ .

Then for all ρ ∈ hJ , the map Rρ : Λ2V → hJ from (4) satisfies Ric(Rρ)(x, y) = 4ω(ρx, y) +ε tr(Jρ)ω(Jx, y) for all x, y ∈ V , and as a hJ -module, we have the deomponsition

K(hJ) = R⊕W,

whereR := Rρ | ρ ∈ hJ ∼= hJ , and W := ker(Ric).

Thus, R decomposes into two irreducible summands, one of which is trivial and spanned by RJ .Moreover, dimW = 1

64n2(n − 2)(n + 6) where n := dimR V . In particular, W 6= 0 iff dimV ≥ 4,

and in this case, W is irreducible as well.

Proof. It is straightforward to calculate that Rρ(ω−1) = 4ρ + ε tr(Jρ)J form which the asserted

formula for the Ricci curvature follows by Lemma 2.1. In particular, Ric(Rρ) 6= 0 for all 0 6= ρ ∈ hJwhich shows the asserted decomposition.

In order to calculate the dimension of W and show that it is irreducible, observe that all the Liealgebras hJ have the same complexification, so it suffices to treat the Lie subalgebra hC := gl(n,C)acting on the vector space V := W ⊕W ∗, where W := C

n is the standard representation.Let x, y ∈W and z,w ∈W ∗. Then for any R ∈ K(hC) we have R(z, x)y−R(z, y)x = −R(x, y)z,

and since the left hand side lies in W while the right hand side lies in W ∗, it follows that both sidesvanish.

The vanishing of the right hand side implies that R(W,W ) = 0 since x, y ∈ W and z ∈ W ∗

are arbitrary. Analogously, R(W ∗,W ∗) = 0. Moreover, the vanishing of the left hand side impliesthat R(z, x)y = R(z, y)x and, analogously, R(x, z)w = R(x,w)z. Thus, if we define the tensorσR ∈W ⊗W ⊗W ∗ ⊗W ∗ by

σR(x, y, z, w) := w(R(z, x)y) = −(R(z, x)w)y for all x, y ∈W and z,w ∈W ∗, (7)

then σR is symmetric in x and y and in z and w, i.e. σR ∈ S2(W ) ⊗ S2(W ∗).Conversely, given σ ∈ S2(W )⊗S2(W ∗), we verify that the map Rσ : Λ2(V ) → h determined by

R(W,W ) = R(W ∗,W ∗) = 0 and (7) lies in K(hC), showing that K(hC) ∼= S2(W ) ⊗ S2(W ∗), andthe decomposition of this space as a hC-module as well as the dimension of this space follow fromstandard representation theoretical arguments.

By virtue of this proposition, we have for each of the Lie algebras hJ = u(p, q) or hJ = u′(m)

K(hJ) ∼= Rp ⊕Wp. (8)

for each p ∈M , and we make the following

Definition 2.8 Let (M,ω, J) be a symplectic manifold with a section J of the endomorphism bun-dle of M such that Jp ∈ sp(TpM,ωp) and J2

p = ε IdTpM for all p ∈M , where ε = ±1.Consider a symplectic connection ∇ for which J is parallel which thus is bi-Lagrangian if ε = 1,

or the Levi-Civita connection of the Kahler metric g := ω(J , ) if ε = −1.If the curvature of ∇ satisfies R∇

p ∈ Rp for all p ∈M with the decomposition (8), then we call∇ Bochner-bi-Lagrangian in the first and Bochner-Kahler in the second case.

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If ∇ is Bochner-Kahler or Bochner-bi-Lagrangian, then there is a unique section ρ of endo-morphisms of the tangent spaces of M for which the curvature of ∇ at each point has the form(4).

For Kahler metrics, the decomposition (8) has first been achieved by Bochner ([Bo]), and forthis reason, the Ricci flat component W of the curvature is called the Bochner curvature of theKahler metric. The same terminology is also adapted for bi-Lagrangian connections.

2.3 Connections with special symplectic holonomy

Let (M,ω) be a symplectic manifold which in this section we assume to be simply connected. If ∇is a symplectic connection on M , then evidently, ω is invariant under parallel translation, so thatthe holonomy group of ∇ is contained in the symplectic group Sp(V, ω). We say that ∇ has specialsymplectic holonomy if the holonomy group of ∇ is contained in an absolutely irreducible propersubgroup of Sp(V, ω). Here, recall that a subgroup H ⊂ Aut(V ) is called absolutely irreducible if Hacts irreducibly on V , and the complexified group HC acts irreducibly on VC := V ⊗ C under thecomplexified representation.

The absolutely irreducible proper subgroups H ⊂ Sp(V, ω) which can occur as the holonomyof a torsion free connection have been classified. The list of these connections is the following (cf.[MS], [S3]).

Table 1: List of Real Special Symplectic Holonomies

Group H Representation space Group H Representation space

SL(2,R) R4 ≃ S3(R2) E57 R56

SL(2,R) · SO(p, q) R2(p+q), (p+ q) ≥ 3 E77 R56

Sp(1)SO(n,H) Hn ≃ R4n, n ≥ 2 Spin(2, 10) R32

SL(6,R) R20 ≃ Λ3

R6 Spin(6, 6) R

32

SU(1, 5) R20 ⊂ Λ3C6 Spin(6,H) R32

SU(3, 3) R20 ⊂ Λ3C6 Sp(3,R) R14 ⊂ Λ3C6

For these representations, the following is known (cf. Proposition 3.3 below).

Proposition 2.9 [MS], [S3] Let H ⊂ Sp(V, ω) be a special symplectic holonomy group with Liealgebra h ⊂ sp(V, ω). Then h is a special symplectic subalgebra, i.e. there is a linear map :S2(V ) → h which satisfies (3).

Thus, by Proposition 2.3, for each ρ ∈ h, the map Rρ : Λ2V → h defined in (4) is contained inK(h). Moreover, we have the following

Proposition 2.10 [MS], [S3] For each special symplectic holonomy group H with Lie algebra h,we have Ric(Rρ) 6= 0 for all ρ 6= 0, and moreover, K(h) = Rρ | ρ ∈ h. Thus, K(h) ∼= h as anH-module.

This means that for each symplectic connection ∇ on (M,ω) with special symplectic holonomy,there is a unique section ρ of endomorphisms of the tangent spaces of M for which the curvatureof ∇ at each point has the form (4).

10

2.4 Special symplectic connections

The various conditions for symplectic connections which we presented in the preceding sectionshave at first glance nothing in common. On the other hand, there is a striking formal similarity inthe presentation of the curvature which turns out to be of great significance. Therefore, we makethe following

Key Definition 2.11 Let (M,ω) be a symplectic manifold of dimension at least 4, equipped witha symplectic connection ∇, i.e. a torsion free connection for which ω is parallel. We say that ∇is a special symplectic connection with structure group H ⊂ Sp(V, ω) and structure Lie algebrah ⊂ sp(V, ω) if

1. ∇ is of Ricci type in the sense of Definition 2.5 In this case, H = Sp(V, ω) and h = sp(V, ω).

2. ∇ is Bochner-Kahler in the sense of Definition 2.8 for a Kahler metric of signature (p, q). Inthis case, H = U(p, q) and h = u(p, q).

3. ∇ is Bochner-bi-Lagrangian in the sense of Definition 2.8. In this case, H ∼= GL(m,R) ⊂Aut((Rm) ⊕ (Rm)∗) and h = u′(m) ⊂ End((Rm) ⊕ (Rm)∗).

4. The holonomy of ∇ is contained in the special symplectic holonomy group H ⊂ Sp(V, ω) withLie algebra h ⊂ sp(V, ω).

Note that in all cases, the structure Lie algebra h ⊂ sp(V, ω) is special symplectic in the senseof Definition 2.2, and there is a section ρ of the endomorphism bundle for which the curvature isgiven by Rρ as in (4).

We are now interested in the covariant derivative of the curvature of a special symplectic con-nection. For this, we consider one of the structure Lie algebras h ⊂ sp(V, ω), and define the spaceof covariant R-derivations by

R(1)h :=

ψ ∈ V ∗ ⊗ h | Rψ(x)(y, z) +Rψ(y)(z, x) +Rψ(z)(x, y) = 0 for all x, y, z ∈ V

.

Again, R(1)h is an H-module in an obvious way. The significance of this space is due to the fact that

for any torsion free connection the second Bianchi identity holds:

(∇xR)(y, z) + (∇yR)(z, x) + (∇zR)(x, y) = 0 for all x, y, z ∈ TpM and p ∈M .

Thus, if ∇ is special symplectic so that at each point p ∈ M the curvature is of the form Rρ forsome ρ ∈ h, then ∇xR is also of this form, so that the correspondence x 7→ ∇xR yields a linear

map ψ : V → h so that ∇xR = Rψ(x). The second Bianchi identity then implies that ψ ∈ R(1)h .

From (3) it is straightforward to verify that for each u ∈ V , the map ψu : V → h, ψu(x) := ux

is contained in R(1)h . Evidently, the correspondence u 7→ ψu is equivariant and injective, so that

R(1)h contains a submodule isomorphic to V . Indeed, this exhausts all of R

(1)h . Namely, we have

the

Proposition 2.12 [CS] If dimV ≥ 4 and h ⊂ sp(V, ω) is one of the subalgebras associated to a

special symplectic connection as in Definition 2.11, then R(1)h

∼= V as an H-module, i.e.

R(1)h = ψu : V → h | u ∈ V , where ψu(x) := u x for all x ∈ V .

11

Now we are ready to investigate the structure equations of special symplectic connections.

Proposition 2.13 [CS] Let (M,ω,∇) be a simply connected symplectic manifold of dimensionat least 4 with a special symplectic connection of regularity C4 with the associated Lie subgroupH ⊂ Sp(V, ω) and special symplectic Lie subalgebra h ⊂ sp(V, ω) from Definition 2.11. Then thereis an associated H-structure π : B → M on M which is compatible with ∇, and there are mapsρ : B → h, u : B → V and f : B → R, such that the tautological form θ ∈ Ω1(B) ⊗ V , theconnection form η ∈ Ω1(B) ⊗ h and the functions ρ, u and f satisfy the structure equations

dθ + η ∧ θ = 0,

dη + 12 [η, η] = Rρ(θ ∧ θ),

dρ+ [η, ρ] = u θ

du+ η · u = (ρ2 + f) · θ

df + d(ρ, ρ) = 0.

(9)

The assumption of simply connectedness of M is imposed only to ensure that the H-structureB →M exists which slightly simplifies the proof. However, our result also hold if M is not simplyconnected.

For clarification, we reformulate the structure equations (9) as follows. If for h ∈ h and x ∈ Vwe let ξh, ξx ∈ X(B) be the vector fields characterized by

θ(ξh) ≡ 0, η(ξh) ≡ h and θ(ξx) ≡ x, η(ξx) ≡ 0,

then (9) holds iff for all h, l ∈ h and x, y ∈ V ,

[ξh, ξl] = ξ[h,l], [ξh, ξx] = ξhx, [ξx, ξy] = −2ω(x, y)ξρ − ξxρy + ξyρx

ξh(ρ) = −[h, ρ], ξh(u) = −hu, ξh(f) = 0,

ξx(ρ) = u x, ξx(u) = (ρ2 + f)x, ξx(f) = −2ω(ρu, x)

Proof. Let F be the H-structure on the manifoldM , and denote the tautological and the connection1-form on F by θ and η, respectively. Since by hypothesis, the curvature maps are all contained inRh, it follows that there is an H-equivariant map ρ : B → h such that the curvature at each pointis given by Rρ with the notation from (4). Thus, we have the structure equations

dθ + η ∧ θ = 0dη + 1

2 [η, η] = Rρ · (θ ∧ θ),(10)

The H-equivariance of ρ yields that ξh(ρ) = −[h, ρ] for all h ∈ h. Moreover, since the covariant

derivative of the curvature is represented by ξx(ρ) for all x ∈ V and this must lie in R(1)h , it follows

by Proposition 2.12 that ξx(ρ) = u ρ for some H-equivariant map u : B → V , which shows theasserted formula

dρ+ [η, ρ] = u θ.

Since u is H-equivariant, it follows that ξh(u) = −hu for all h ∈ h. Also, elaborating theequation ξxξyρ− ξyξxρ = [ξx, ξy]ρ yields that for all x, y ∈ V

(

ξxu− ρ2x)

y =(

ξyu− ρ2y)

x. (11)

Now we need the following lemma whose proof we will postpone.

12

Lemma 2.14 Let h ⊂ sp(V, ω) be a special symplectic subalgebra, dimV ≥ 4, and let ϕ : V → Vbe a linear map such that

ϕ(x) y = ϕ(y) x for all x, y ∈ V . (12)

Then ϕ is a multiple of the identity.

Applying the lemma to the function x 7→ ξxu−ρ2x, (11) implies that there is a smooth function

f : B → R for which ξxu− ρ2x = fx for all x ∈ V so that

du+ η · u = (ρ2 + f)θ.

Finally, elaborating the equation ξxξyu− ξyξxu = [ξx, ξy]u yields that df + d(ρ, ρ) = 0.

Proof of Lemma 2.14. By (3) we have

(ϕ(x) y)z − (ϕ(x) z)y = 2ω(y, z)ϕ(x) + ω(ϕ(x), z)y − ω(ϕ(x), y)z.

But (12) now implies that the cyclic sum in x, y, z of the left hand side vanishes, hence so does thecyclic sum of the right hand side, i.e.

2 (ω(x, y)ϕ(z) + ω(y, z)ϕ(x) + ω(z, x)ϕ(y))

= (ω(ϕ(y), z) − ω(ϕ(z), y))x + (ω(ϕ(z), x) − ω(ϕ(x), z))y + (ω(ϕ(x), y) − ω(ϕ(y), x))z.(13)

For each x ∈ V , we may choose vectors y, z ∈ V with ω(x, y) = ω(x, z) = 0 and ω(y, z) 6= 0 sincedimV ≥ 4. Then (13) implies that ϕ(x) ∈ span(x, y, z) so that ω(ϕ(x), x) = 0. Polarization thenimplies that ω(ϕ(x), y) + ω(ϕ(y), x) = 0 for all x, y ∈ V .

Next, we take the symplectic form of (13) with x, and together with the preceding identity thisyields

ω(x, y)ω(ϕ(x), z) = ω(x, z)ω(ϕ(x), y) for all x, y, z ∈ V .

Thus, ω(x, y)ϕ(x) = ω(ϕ(x), y)x for all x, y ∈ V , and since for 0 6= x ∈ V we can pick y ∈ Vsuch that ω(x, y) 6= 0, this implies that ϕ(x) is a scalar multiple of x for all x ∈ V , whence ϕ is a

multiple of the identity.

It is now our aim to interpret the structure equations (9) in a way that links them to paraboliccontact geometry. This shall be pursued in the following section.

3 Parabolic contact structures

3.1 Two-gradable Lie algebras

Definition 3.1 A real simple Lie algebra g is called 2-gradable if there exists a decomposition

g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 (14)

with dim g±2 = 1, and an element H0 ∈ [g2, g−2] such that adH0 |gi = i Idgi for all i.

13

Since gi is an Eigenspace of adH0 , it follows that [gi, gj ] ⊂ gi+j by the Jacobi identity. Moreover,note that sl0 := g2 ⊕ g−2 ⊕ RH0 is a Lie subalgebra of g which is isomorphic to sl(2,R).

Recall that the complexification gC := g ⊗ C of g is a complex simple Lie algebra for which wecan choose the Cartan decomposition

gC = t ⊕⊕

α∈∆

gα,

where t ⊂ gC is a maximal abelian self-normalizing subalgebra, and ∆ ⊂ t∗\0 is the root system ofgC such that adt|gα = α(t)Idgα for all t ∈ t and α ∈ ∆. Moreover,

g = (g ∩ t) ⊕⊕

α∈∆+

g ∩ (gα ⊕ g−α).

If g is two-gradable then H0 ∈ g is diagonalizable, so that we can choose the Cartan decompo-sition of gC such that H0 ∈ t ∩ g, hence α(H0) ∈ 0,±1,±2 for all α ∈ ∆, and there is exactlyone α0 ∈ ∆ such that g±2 = g ∩ g±α0 and H0 ∈ [gα0 , g−α0 ] is uniquely determined by α0(H0) = 2.Thus, we have

gi =⊕

β∈∆|〈β,α0〉=i

gβ for i 6= 0, and g0 = t ⊕⊕

β∈∆|〈β,α0〉=0

gβ.

Here, 〈β, α〉 deontes the Cartan number. In particular, g±2 = g±α0 , and if ∆ has roots of differentlength, then α0 must be a long root, as | 〈β, α0〉 | ≤ 1 for all β 6= ±α0.

We can decompose g0 = RHα0 ⊕ h, where the Lie algebra h is characterized by [h, sl0] = 0.Observe that g0 and hence h are reductive. Thus, as a Lie algebra,

gev := g−2 ⊕ g0 ⊕ g2 ∼= sl0 ⊕ h and godd := g−1 ⊕ g1 ∼= R2 ⊗ V as a gev-module,

where h acts effectively on V . Identifying h with its image under this representation, we may regardit as a subalgebra h ⊂ End(V ), and hence we have the decomposition

g = gev ⊕ godd ∼= (sl(2,R) ⊕ h) ⊕ (R2 ⊗ V ), (15)

where this notation indicates the representation ad : gev → End(godd).We fix a non-zero R-bilinear area form a ∈ Λ2(R2)∗. There is a canonical sl(2,R)-equivariant

isomorphim

S2(R2) −→ sl(2,R), (ef) · g := a(e, g)f + a(f, g)e for all e, f, g ∈ R2, (16)

and under this isomorphism, the Lie bracket on sl(2,R) is given by

[ef, gh] = a(e, g)fh + a(e, h)fg + a(f, g)eh + a(f, h)eg.

Thus, if we fix a basis e+, e− ∈ R2 with a(e+, e−) = 1, then we have the identifications

H0 = −e+e−, g±2 = Re2±, g±1 = e± ⊗ V.

Now one can show the following

14

Proposition 3.2 [CS] Let g be a 2-gradable simple Lie algebra, and consider the decompositions(14) and (15). Then there is an h-invariant symplectic form ω ∈ Λ2V ∗ and an h-equivariant product : S2(V ) → h such that

[ , ] : Λ2(godd) −→ gev ∼= sl(2,R) ⊕ h

is given as

[e⊗ x, f ⊗ y] = ω(x, y)ef + a(e, f)x y for e, f ∈ R2 and x, y ∈ V , (17)

using the identification S2(R2) ∼= sl(2,R) ⊂ gev from (16). Moreover, there is a multiple ( , ) ofthe Killing form on g which satisfies the following:

1. (gi, gj) = 0 if i+ j 6= 0,

2. For all x, y ∈ V and h ∈ h, we have (h, x y) = ω(hx, y) = ω(hy, x)

3. For all x, y, z ∈ V , (3) holds, so that h ⊂ sp(V, ω) is a special symplectic subalgebra.

Thus, each 2-gradable simple Lie algebra yields a special symplectic subalgebra h ⊂ sp(V, ω).The converse is also true. Namely, we have

Proposition 3.3 [CS] Let (V, ω) be a symplectic vector space, and let h ⊂ sp(V, ω) be a specialsymplectic subalgebra with product : S2(V ) → h. Then there is a unique 2-gradable simple Liealgebra g which admits the decompositions (14) and (15), and the Lie bracket of g is given by (17).

We shall call g the simple Lie algebra associated to the special symplectic subalgebra h ⊂ sp(V, ω).From this proposition, we obtain a complete classification of special symplectic subalgebras

by considering all 2-gradable real simple Lie algebras ([OV]). Namely, a simple Lie algebra g is2-gradable iff we can choose the Cartan decomposition of gC such that g ∩ g±α0 6= 0 for some longroot α0 ∈ ∆.

Corollary 3.4 Table 2 yields the complete list of special symplectic subgroups H ⊂ Sp(V, ω).

From Table 2, we now observe the link between 2-gradable simple Lie algebras and the Lie groupsH ⊂ Sp(V, ω) which are associated to special symplectic connections in the sense of Definition 2.11.Namely, note that the Lie groups H ⊂ Sp(V, ω) corresponding to entries (i) and (ii) are preciselythe groups associated to Bochner-bi-Lagrangian and Bochner Kahler connections; the H of entry(iii) is associated to the connections of Ricci type, whereas a comparison with Table 1 yields thatthe H’s of entries (iv) – (xv) are the special symplectic holonomy groups.

This shall be the conceptual background of our construction of special symplectic connectionsout of 2-gradable simple Lie algebras.

3.2 Contact manifolds

We shall now recall some well known facts about contact manifolds and their symplectic reductions.

Definition 3.5 A contact structure on a manifold C is a smooth distribution D ⊂ TC of codimen-sion one such that the Lie bracket induces a non-degenerate map

D ×D −→ TC/D =: L.

15

Table 2: Real 2-gradable Lie groups

Type of ∆ G H V

(i) Ak, k ≥ 2 SL(n + 2, R), n ≥ 1 GL(n, R) W ⊕ W ∗ with W ∼= Rn

(ii) SU(p + 1, q + 1), p + q ≥ 1 U(p, q) Cp+q

(iii) Ck, k ≥ 2 Sp(n + 1, R) Sp(n, R) R2n

(iv) Bk, k ≥ 3 SO(p + 2, q + 2), p + q ≥ 3 SL(2, R) · SO(p, q) R2⊗ R

p+q

(v) Dk, k ≥ 4 SO(n + 2, H), n ≥ 2 Sp(1) · SO(n, H) Hn

(vi) G2 G′

2 SL(2, R) S3(R2)

(vii) F4 F(1)4 Sp(3, R) R

14⊂ Λ3

R6

(viii) E6 ER6 SL(6, R) Λ3

R6

(ix) E(2)6 SU(1, 5) R

20⊂ Λ3

C6

(x) E(3)6 SU(3, 3) R

20⊂ Λ3

C6

(xi) E7 E(5)7 Spin(6, 6) R

32⊂ ∆C

(xii) E(6)7 Spin(6, H) R

32⊂ ∆C

(xiii) E(7)7 Spin(2, 10) R

32⊂ ∆C

(xiv) E8 E(8)8 E

(5)7 R

56

(xv) E(9)8 E

(7)7 R

56

The line bundle L→ C is called the contact line bundle, and its dual can be embedded as

L∗ = λ ∈ T ∗C | λ(D) = 0 ⊂ T ∗C. (18)

Notice that we can define the line bundles L → C and L∗ → C for an arbitrary distributionD ⊂ TC of codimension one. It is well known that such a distribution D yields a contact structureiff the restriction of the canonical symplectic form Ω on T ∗C to L∗\0 is non-degenerate, so that inthis case L∗\0 is a symplectic manifold in a canonical way.

We regard p : L∗\0 → C as a principal (R\0)-bundle. We call the contact structure orientable ifL∗\0 has two components each of which is a principal R

+-bundle. In fact, if the contact structureis not orientable, then there is a double cover C of C such that the induced contact structure D onC is orientable. Thus, for an orientable contact structure, we get the principal R

+-bundle

p : C −→ C,

where C ⊂ L∗\0 is a connected component. The choice of connected component is called anorientation of the contact structure.

The vector field E0 ∈ X(C) which generates the principal action is called Euler field, so that theflow along E0 is fiberwise scalar multiplication in C ⊂ L∗ ⊂ T ∗C. Thus, the Liouville form on T ∗Cis given as λ := E0 Ω, and hence LE0(Ω) = Ω and Ω = dλ. This process can be reverted. Namely,we have the following

Proposition 3.6 Let p : C → C be a principal R+-bundle with a symplectic form Ω on C such

that LE0Ω = Ω where E0 ∈ X(C) generates the principal action. Then there is a unique contactstructure D on C and an equivariant imbedding ı : C → L∗\0 ⊂ T ∗C with L∗ from (18) such that Ωis the pullback of the canonical symplectic form on T ∗C to C.

16

Proof. By hypothesis, Ω = dλ where λ := (E0 Ω). Since λ(E0) = 0, there is for each x ∈ C aunique λx ∈ T ∗

p(x)C satisfying p∗(λx) = λx. Moreover, LE0(λ) = λ, hence λetx = etλx for all t ∈ R, so

that the codimension one distribution D := dp(ker(λ)) ⊂ TC is well defined, and the correspondencex 7→ λx yields an equivariant imbedding C → L∗\0 whose image is thus a connected componentof L∗\0. Moreover, by construction, λ is the restriction of the Liouville form to C ⊂ L∗\0 ⊂ T ∗C.

Since Ω = dλ is non-degenerate on C by assumption, it follows that D is a contact structure.

Next, we define the fiber bundle

R := (λ, ξ) ∈ C × T C ⊂ T ∗C × T C | λ(dp(ξ)) = 1.

Projection onto the first factor yields a fibration R → C whose fiber is an affine space.

Definition 3.7 Let C be a contact manifold. We call a vector field ξ on C a contact symmetry ifLξ(D) ⊂ D. This means that the flow along ξ preserves the contact structure D.

We call ξ a transversal contact symmetry if in addition ξ 6∈ D at all points. If C is orientedwith orientation C ⊂ L∗\0, then ξ is called positively transversal if λ(ξ) > 0 for all λ ∈ C.

For each contact symmetry ξ on C, there is a unique vector field ξ ∈ X(C), called the Hamiltonianlift of ξ, satisfying dp(ξ) = ξ and L

ξλ = 0, so that L

ξΩ = 0.

Given a positively transversal contact symmetry ξ with Hamiltonian lift ξ, there is a uniquesection λ of the bundle p : C → C such that λ(ξ) ≡ 1, and hence we obtain a section of the bundleR → C → C

σξ : C −→ R, σξ := (λ, ξ) ∈ R. (19)

We call an open subset U ⊂ C regular w.r.t. the transversal contact symmetry ξ if there is asubmersion πU : U → MU onto some manifold MU whose fibers are connected lines tangent to ξ.Evidently, since ξ is pointwise non-vanishing, C can be covered by regular open subsets.

Since ξ is a contact symmetry, it follows that ξ dλ = 0 and Lξλ = 0. Thus, on each MU thereis a unique symplectic form ω such that

π∗Uω = −2dλ, (20)

where the factor −2 only occurs to make this form coincide with one we shall construct later on.

3.3 Parabolic contact structures

To link all of this to our situation, let g be a 2-gradable simple real Lie algebra and let G be thecorresponding connected Lie group with trivial center Z(G) = 1. Recall the decomposition

g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2 ∼= Re2− ⊕ e− ⊗ V ⊕ (Re+e− ⊕ h) ⊕ e+ ⊗ V ⊕ Re2+

from (14). We let µ := g−1dg be the left invariant Maurer-Cartan form on G, which we candecompose as

µ =2

∑

i=−2

µi, µ0 = µh + ν0e+e− (21)

17

where µi ∈ Ω1(G) ⊗ gi, µh ∈ Ω1(G) ⊗ h and ν0 ∈ Ω1(G). Furthermore, we define the subalgebras

p := g0 ⊕ g1 ⊕ g2, and p0 := h ⊕ g1 ⊕ g2,

and we let P,P0 ⊂ G be the corresponding connected subgroups. Using the bilinear form ( , ) fromProposition 3.2, we identify g and g∗. Now we define the root cone and its (oriented) projectivization

C := G · e2+ ⊂ g ∼= g∗, C := p(C) ⊂ Po(g) ∼= P

o(g∗),

where Po(g) is the set of oriented lines in g, i.e. P

o ∼= Sdimg−1, where p : g\0 → Po(g) is the principal

R+-bundle defined by the canonical projection. Thus, the restriction p : C → C is a principal bundle

as well.Being a coadjoint orbit, C carries a canonical G-invariant symplectic structure Ω. Moreover,

the Euler vector field defined by

E0 ∈ X(C), (E0)v := v

generates the principal action of p and satisfies LE0(Ω) = Ω, so that the distribution D = dp(E⊥Ω0 ) ⊂

TC yields a G-invariant contact distribution on C by Proposition 3.6. Now one can show thefollowing

Lemma 3.8 [CS] As homogeneous spaces, we have C = G/P, C = G/P0 and R = G/H. Moreover,the fiber bundles R → C → C from before are equivalent to the corresponding homogeneous fibrations.

For each a ∈ g we define the vector fields a∗ ∈ X(C) and a∗ ∈ X(C) corresponding to theinfinitesimal action of a, i.e.

(a∗)[v] :=d

dt

∣

∣

∣

∣

t=0

(exp(ta) · [v]) and (a∗)v :=d

dt

∣

∣

∣

∣

t=0

(exp(ta) · v). (22)

Note that a∗ is a contact symmetry and a∗ is its Hamiltonian lift. Let

Ca := λ ∈ C | λ(a∗) > 0 and Ca := p(Ca) ⊂ C, (23)

so that p : Ca → Ca is a principal R+-bundle and the restriction of a∗ to Ca is a positively transversal

contact symmetry. Therefore, we obtain the section σa : Ca → R = G/H from (19).Let π : G → G/H = R be the canonical projection, and let Γa := π−1(σa(Ca)) ⊂ G. Then

evidently, the restriction π : Γa → σa(Ca) ∼= Ca is a (right) principal H-bundle.

Theorem 3.9 [CS] Let a ∈ g be such that Ca ⊂ C from (23) is non-empty, define a∗ ∈ X(C) anda∗ ∈ X(C) as in (22), and let π : Γa → Ca with Γa ⊂ G be the principal H-bundle from above. Thenthere are functions ρ : Γa → h, u : Γa → V , f : Γa → R such that

Adg−1(a) =1

2e2− + ρ+ e+ ⊗ u+

1

2fe2+ (24)

for all g ∈ Γa. Moreover, the restriction of the components µh + µ−1 + µ−2 of the Maurer-Cartanform (21) to Γa yields a pointwise linear isomorphism TΓa → h ⊕ g−1 ⊕ g−2, and if we decomposethis coframe as

µh + µ−1 + µ−2 = −2κ(

12e

2− + ρ

)

+ e− ⊗ θ + η,

where κ ∈ Ω1(Γa), θ ∈ Ω1(Γa) ⊗ V, η ∈ Ω1(Γa) ⊗ h,(25)

18

then κ = −12π

∗(λ) where λ ∈ Ω1(Ca) is the contact form for which σa = (λ, a∗). Moreover,

dκ =1

2ω(θ ∧ θ), (26)

and θ, η, ρ, u, f satisfy the structure equations (9).

Proof. According to the above identifications, we have g ∈ Γa iff (g ·e2+, g ·(12e

2−+p0)) = σa([g ·e

2+])

iff g · (12e

2− + p0) = (a∗)g·e2+ iff (Adg−1(a∗))e2+ = 1

2e2− mod p0 iff Adg−1(a) = 1

2e2− mod p0, i.e.

Γa =

g ∈ G∣

∣ Adg−1(a) ∈ Q

,where

Q := 12e

2− + p0 =

12e

2− + ρ+ e+ ⊗ u+ 1

2fe2+ | ρ ∈ h, u ∈ V, f ∈ R

,(27)

and from this (24) follows. Thus, if dLgv ∈ TgΓa with v ∈ g, then we must have

p0 ∋d

dt

∣

∣

∣

∣

t=0

(

Ad(g exp(tv))−1 (a))

= −[v,Adg−1(a)] = −

[

v,1

2e2− + ρ+ e+ ⊗ u+

1

2fe2+

]

,

and from here it follows by a straightforward calculation that v must be contained in the space

RAdg−1a⊕

e− ⊗ x+ e+ ⊗ ρx+1

2ω(u, x)e2+

∣

∣

∣

∣

x ∈ V

⊕ h, (28)

and since v was arbitrary, it follows that µ(TgΓa) is contained in (28). In fact, a dimension countyields that dim(µ(TgΓa)) = dimΓa = dim Ca + dim H coincides with the dimension of (28), hence(28) equals µ(TgΓ), i.e. µh + µ−1 + µ−2 : TΓa → h ⊕ g−1 ⊕ g−2 yields a pointwise isomorphism.From there, the structure equations (26) and (9) follow by a straightforward calculation.

With these equations, it follows that κ is H-invariant and vanishes along the principal fibers,hence κ = −1

2π∗(λ) for some λ ∈ Ω1(Ca). Since κ|µ−1(g−1) = 0, it follows that λ is a contact form.

Moreover, if we let a∗ denote the right invariant vector field on G characterized by µ(a∗) = Adg−1(a),

then dp(a∗) = a∗, where p : Γa → C is the canonical projection, and from (24) it follows that

λ(a∗) = −2κ(a∗) ≡ 1, so that (λ, a∗) ∈ R which shows the final assertion.

Consider the principal Ta-bundle Γa → Ta\Γa =: Ba whose fundamental vector field we denoteby ξa. That is, ξa is the restriction of the right invariant vector field on G corresponding to a ∈ g

to Γa ⊂ G. Thus, µ(ξa) = Adg−1a, and the flow along ξa preserves µ. Therefore, by (21), (24) and(25), it follows that

κ(ξa) ≡ −1

2, η(ξa) ≡ 0, θ(ξa) ≡ 0,

and we obtain the following

Corollary 3.10 1. The differential form κ ∈ Ω1(Γa) defined above yields a connection on theprincipal Ta-bundle p : Γa → Ba from above.

2. The functions ρ : Γa → h, u : Γa → V and f : Γa → R defined above are constant along thefibers of p : Γa → Ba, hence they induce functions on Ba which by abuse of notation shallalso be denoted by ρ, u and f , respectively.

19

3. There are differential forms on Ba whose pull back under p equals θ and η, respectively. Byabuse of notation, these forms will also be denoted by θ and η, respectively.

4. θ + η ∈ Ω1(Ba) ⊗ (V ⊕ h) is a coframing, i.e. yields a pointwise isomorphism of TBa withV ⊕ h, and θ, η, ρ, u and f satisfy the structure equations (9).

5. There is a differential form on Ca = Γa/H whose pull back equals κ, and again, we shalldenote this form also by κ.

Thus, if we let Xa := Ta\Γa/H, then we obtain the following commutative diagram, where thelabeled maps are principal bundles with the indicated structure groups.

ΓaTa //

H

Ba

Ca // Xa

(29)

Let us assume for the moment that Xa is a manifold. Then the maps Ba → Xa and Ca → Xa

are principal bundles with structure group H and Ta, respectively. Moreover, κ yields a connectionon the Ta-bundle Ca → Xa, and θ induces an embedding of Ba into the coframe bundle of Xa suchthat θ is the pull back of the tautological form. Hence we may regard Ba → Xa as an H-structureon Xa, and η yields a connection on this structure which satisfies (9) by Corollary 3.10 and hence isspecial symplectic. In particular, Xa carries a symplectic structure ω which by (26) is the curvatureof the connection κ on the line bundle Ca → Xa. Therefore, this line bundle is a quantization bundleof Xa.

If Ta∼= S1, then Xa is an orbifold, and the statements in the preceding paragraph are still valid

in this sense, i.e. Xa carries a special symplectic orbifold connection and Ca → Xa is the orbifoldquantization bundle.

However, for general a ∈ g, Xa will be neither a manifold nor an orbifold. In fact, there areplenty of choices where Xa is neither Hausdorff nor locally Euclidean, so the statement that Xa

should carry a special symplectic connection is somewhat delicate. Therefore, in order to get aprecise statement in the general case, we have to “localize” the structure.

For this, note that the action of Ta on Ca whose quotient equals Xa is locally free. Thus, we cancover Ca = Γa/H by open subsets U ⊂ Ca with the property that the local quotient MU := Tloc

a \U ,i.e. the set of connected components of the intersections of Ta-orbits with U , is a manifold. We letΓU := p−1(U) ⊂ Γa where p : Γa → Ca is the principal H-bundle from above. Thus, restricting themaps of (29), we get the following commutative diagram

ΓUTloc

a //

H

BlocU

H

UTloc

a // MU

where BlocU := Tloc

a \ΓU . Now we argue again that the differential forms θ and η on ΓU ⊂ Γaas well as the functions ρ, u and f fator through to differential forms and functions on Bloc

U ,respectively, which satisfy (9). Thus, we may regard Bloc

U →MU as an H-structure, and η yields aspecial symplectic connection on MU . Moreover, we can extend the local principal bundle U →MU

20

to a principal Ta-bundle U → MU and we can extend κ ∈ Ω1(U) to a connection form on U forwhich (26) holds. Thus, U → MU is a quantization bundle of MU . That is, we have the followingresult from which Theorem A from the introduction follows immediately.

Theorem 3.11 Let g be a 2-gradable real simple Lie algebra, and let h ⊂ g be the Lie subalgebrafrom (15), i.e. h ⊂ sp(V, ω) is special symplectic. Let a ∈ g and Ca ⊂ C as before. Let U ⊂ Ca bean open subset for which the local quotient MU := Tloc

a \U is a manifold, where

Ta := exp(Ra) ⊂ G.

Let ω ∈ Ω2(MU ) be the symplectic form from (20). Then MU carries a canonical special symplecticconnection associated to h, and the (local) principal Ta-bundle π : U → MU admits a connectionκ ∈ Ω1(U) whose curvature is given by dκ = π∗(ω).

Remark 3.12 If we replace a by a′ := Adg0(a), then it is clear that in the above construction wehave Γa′ = Lg0Γa. Thus, identifying Γa and Γa′ via Lg0 , the functions ρ + µ + f and the formsκ+ θ+ω will be canonically identified and hence both satisfy (9). Therefore, the connections fromthe preceding theorem only depend on the adjoint orbit of a.

Also, since Ca = Cet0a and Ta = Tet0a for all t0 ∈ R, the above construction yields equivalentconnections when replacing a by et0a. In this case, however, the symplectic form ω on the quotientwill be replaced by e−t0ω.

4 The developing map

In this section, we shall revert the process of the preceding section, showing that any specialsymplectic connection is equivalent to the ones given in Theorem 3.11 in a sense which is to be madeprecise. Namely, recall that by Proposition 2.13 each special symplectic connection of regularityC4 associated to the special symplectic Lie algebra h ⊂ sp(V, ω) on a symplectic manifold (M,ω)of dimension at least 4 induces functions ρ : B → h, u : B → V and f : B → R where π : B → Mis the associated H-structure on M , such that for the tautological one form θ ∈ Ω1(B)⊗V and theconnection form η ∈ Ω1(B) ⊗ h, the structure equations (9) hold.

It is now our aim to construct the equivalent to the principal line bundle Γ → B from thepreceding section. Namely, we let g be the 2-gradable simple Lie algebra associated to h by Propo-sition 3.3. Motivated by (27) and (28), we define the following function A and one form σ

A : B −→ Q ⊂ g, A := 12e

2− + ρ+ e+ ⊗ u+ 1

2fe2+,

σ ∈ Ω(B) ⊗ g, σ := e− ⊗ θ + η + e+ ⊗ (ρθ) + 12ω(u, θ)e2+,

(30)

where Q := 12e

2− + p0 ⊂ g is the affine hyperplane from (27). It is then straightforward to verify

that (9) is equivalent to

dA = −[σ,A] and dσ +1

2[σ, σ] = 2π∗(ω)A. (31)

Let us now enlarge the principal H-bundle B →M to the principal G-bundle

B := B ×H G −→M,

21

where H acts on B × G from the right by (b, g) · h := (b · h, h−1g), using the principal H-action onB in the first component. Evidently, the inclusion B×H → B×G induces an embedding B → B.

Proposition 4.1 The function A and the one form α defined by

A : B −→ g, A([b, g]) := Adg−1(A(b)),

α ∈ Ω1(B) ⊗ g, α[(b,g)] := Adg−1σb + µ,

on B are well defined, where µ = g−1dg ∈ Ω1(G) ⊗ g is the left invariant Maurer-Cartan form onG, and the restriction of A to B ⊂ B coincides with A. Moreover, α yields a connection on theprincipal G-bundle B →M which satisfies

dA = −[α,A] and dα+1

2[α,α] = 2π∗(ω)A. (32)

Proof. First, note that A : B → H and σ ∈ Ω1(B) ⊗ g are H-equivariant, i.e. R∗hA = Adh−1A and

R∗hσ = Adh−1σ. Thus, if we define the function A and the one form α by

A := Adg−1(A) : B × G −→ g

α := Adg−1σ + µ ∈ Ω1(B × G) ⊗ g,

then A(bh, h−1g) = A(b, g), so that A is the pull back of a well defined function A : B → g. Also,α is invariant under the right H-action from above, and for h ∈ h we have

α((ξh)b, dRg(−h)) = Adg−1(σb(ξh)) − µ(dRg(h)) = Adg−1(h) − Adg−1(h) = 0,

so that α is indeed the pull back of a well defined form α ∈ Ω1(B)⊗ g. Moreover, R∗g(α) = Ad−1

g αis easily verified, and since α coincides with µ on the fibers of the projection B×G → B, it followsthat the value of α on each left invariant vector field on G is constant. Since the left invariantvector fields generate the principal right action of the bundle B × G → B, it follows that α is aconnection on this bundle, hence so is α on the quotient B →M .

Finally, to show (32) it suffices to show the corresponding equations for α and A. We have

dA = −[µ,Adg−1(A)] + Adg−1(dA) = −[µ, A] − Adg−1([σ,A]) = −[µ, A] − [Adg−1σ, A] = −[α, A]

by (31), and

dα+ 12 [α, α] = (−[µ,Adg−1σ] + Adg−1dσ + dµ) + 1

2(Adg−1 [σ, σ] + 2[µ,Adg−1σ] + [µ, µ])

= Adg−1(dσ + 12 [σ, σ]) + dµ + 1

2 [µ, µ]

= Adg−1(2π∗(ω)A) = 2π∗(ω)A,

where the second to last equation follows from the Maurer-Cartan equation and (31).

Let M ⊂ B be a holonomy reduction of α, and let T ⊂ G be the holonomy group, so that therestriction M → M becomes a principal T-bundle. By the first equation of (32), it follows that

22

M ⊂ A−1(a) for some a ∈ g, and by choosing the holonomy reduction such that it contains anelement of B ⊂ B, we may assume w.l.o.g. that a ∈ Q. We let

S := Stab(a) = g ∈ G | Adga = a ⊂ G and s := z(a) = x ∈ g | [x, a] = 0,

so that S ⊂ G is a closed Lie subgroup whose Lie algebra equals s. Observe that the restrictionA−1(a) →M is a principal S-bundle, hence we conclude that T ⊂ S. Moreover, on M , we have

α = 2κa

for some κ ∈ Ω1(M) which by (32) satisfies dκ = π∗(ω). In particular, the Ambrose-SingerHolonomy theorem implies that Ta = exp(Ra) ⊂ G is the identity component of T which is thus aone dimensional (possibly non-regular) subgroup of S, and κ yields the desired connection form onthe principal T-bundle M →M .

Define Ca ⊂ C as in (23) and Γa ⊂ G and Q ⊂ g as in (27), and let

B := p−1(M ) ⊂ B × G,

where p : B × G → B ×H G = B is the canonical projection. Then the restriction of the map

ı : B × G −→ G, ı(b, g) := g−1

satisfies ı(B) ⊂ Γa; indeed, since A(M) ≡ a, it follows that Adg−1A(b) = a for all (b, g) ∈ B andhence Adga = A(b) ∈ Q, so that g−1 ∈ Γa. Since 2κa = α = Adg−1σ + µ, it follows by (30) that

ı∗(µ) = −Adgµ = −2κAdga+ σ = −2κA+ e− ⊗ θ + η + e+ ⊗ (ρθ) +1

2ω(u, θ)e2+,

and hence

ı∗(µ) = −2κ

(

1

2e2− + ρ

)

+ e− ⊗ θ + η mod g1 ⊕ g2.

Comparing this equation with the structure equations in Theorem 3.9, it follows that the inducedmap ı : M = B/H → Ca = Γa/H is a local diffeomorphism and the induced map ı : M := T\M →Ta\Ca is connection preserving, where Ta\Ca is (locally) equipped with the special symplecticconnection from Theorem 3.11. Thus, we have shown Theorem B from the introduction.

Remark 4.2 Theorem B generalizes immediately to orbifolds. Namely, if M is an orbifold, then aspecial symplectic orbifold connection consists of an almost principal H-bundle B →M , i.e. H actslocally freely and properly on B such that M = B/H, and a coframing θ + η ∈ Ω1(B) ⊗ (V ⊕ h)on B such that η(ξh) ≡ h ∈ h and θ(ξh) ≡ 0 for all infinitesimal generators ξh of the H-action, andsuch that the structure equations (10) hold for some function ρ : B → h.

Now the proofs of Propositions 2.13 and 4.1 as well as the proof of Theorem B go throughverbatim as we never used the freeness of the H-action on B. In particular, the holonomy reductionM is a manifold on which T acts locally freely, and M = T\M as an orbifold.

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5 Applications and global properties

Definition 5.1 Let (M,∇) be a manifold with a connection. A (local) symmetry of the connectionis a (local) diffeomorphism ϕ : M →M which preserves ∇, i.e. such that ∇dϕ(X)dϕ(Y ) = dϕ(∇XY )for all vector fields X,Y on M . An infinitesimal symmetry of the connection is a vector field ζ onM such that for all vector fields X,Y on M we have the relation

[ζ,∇XY ] = ∇[ζ,X]Y + ∇X [ζ, Y ].

Furthermore, let π : B → M be an H-structure compatible with ∇, and let θ, η denote thetautological and the connection form on B, respectively. A (local) symmetry on B is a (local)diffeomorphism ϕ : B → B such that ϕ∗(θ) = θ and ϕ∗(η) = η. An infinitesimal symmetry on Bis a vector field ζ on B such that Lζ(θ) = Lζ(η) = 0.

The ambiguity of the terminology above is justified by the one-to-one correspondence between(local or infinitesimal) symmetries on M and B. Namely, if ϕ : M → M is a (local) symmetry,then there is a unique (local) symmetry ϕ : B → B with π ϕ = ϕ π, and vice versa. Likewise,for any infinitesimal symmetry ζ on M , there is a unique infinitesimal symmetry ζ on B such thatζ = dπ(ζ).

The infinitesimal symmetries form the Lie algebra of the (local) group of (local) symmetries.We also observe that an infinitesimal symmetry on B is uniquely determined by its value at anypoint. (The corresponding statement fails for infinitesimal symmetries on M in general.)

Proof of Corollary C. The first part follows immediately from Theorem B since Ca ⊂ C is anopen subset of the analytic manifold C, and the action of Ta on Ca is analytic as well. Also, theC4-germ of the connection at a point determines uniquely the G-orbit of a ∈ g by (9) and hencethe connection by Theorem B.

Note that the generic element a ∈ g is G-conjugate to an element in the Cartan subalgebrawhich is uniquely determined up to the action of the (finite) Weyl group. Since multiplying a ∈ g

by a scalar does not change the connection, it follows that the generic special symplectic connectionassociated to g depends on (rk(g) − 1) parameters.

For the second part, by virtue of Theorem B it suffices to show the statement for manifolds ofthe form M = MU where U ⊂ Ca is a regular open subset for some a ∈ g. Let ΓU ⊂ Γa ⊂ G bethe H-invariant subset such that we have the principal H-bundle ΓU → U , and let BU := Ta\ΓUso that BU →MU is the associated H-structure.

Let x ∈ s, and denote by ζx the right invariant vector field on G corresponding to −x, so thatthe map x 7→ ζx is a Lie algebra homomorphism. Then L

ζx(µ) = 0 where µ denotes the Maurer-

Cartan form. By (27), it follows that the restriction of ζx to Γa is tangent, and since ΓU ⊂ Γa isopen, we may regard ζx as a vector field on ΓU . Since ζx commutes with the action of Ta, it followsthat there is a related vector field ζx on the quotient BU = Tloc

a \ΓU , and since the tautological andcurvature form of the induced connection on BU pull back to components of µ, it follows that ζx isan infinitesimal symmetry on BU .

Conversely, suppose that ζ is an infinitesimal symmetry on BU . Since an infinitesimal symmetrymust preserve the curvature and its covariant derivatives, we must have ζ(A) = 0. But the tangentof the fiber of the map A : ΓU → g is spanned by the vector fields ζx, x ∈ s, and since infinitesimalsymmetries are uniquely determined by their value at a point, it follows that ζ = ζx for some x ∈ s.

24

Finally, it is evident that ζx = 0 iff ζx is tangent to Ta iff x ∈ Ra, hence the claim follows.

We also mention the following rigidity result from [CS].

Theorem 5.2 Let g be a 2-gradable simple Lie algebra, let G be the connected Lie group with Liealgebra g and trivial center, and let S ⊂ G be a maximal compact subgroup. Then C = S/K forsome compact subgroup K ⊂ S where C ⊂ P

o(g) is the root cone. Moreover, let T ⊂ S be the identitycomponent of the center of S. Then the following are equivalent:

1. There is a compact simply connected symplectic manifold M with a special symplectic con-nection associated to the simple Lie algebra g.

2. dim T = 1, i.e. T ∼= S1.

3. T 6= e.

If these conditions hold then T\C ∼= S/(T ·K) is a compact hermitian symmetric space, and themap ı : M → T\C from Theorem B is a connection preserving covering. Thus, M is a hermitiansymmetric space as well.

This theorem allows us to classify all compact simply connected manifolds with special sym-plectic connections, as the maximal compact subgroups of semisimple Lie groups are fully classified(e.g. [OV]). Thus, we obtain Theorem D from the introduction.

We shall only sketch the proof of Theorem 5.2. If M is simply connected, then there is aprincipal Ta-bundle π : M → M by Theorem B with a connection κ whose curvature equals ω. IfTa

∼= R, then π would be a homotopy equivalence, and since π∗(ω) = dκ is exact, this would implythat ω ∈ Ω2(M) was exact, which is impossible if M is compact.

We conclude that Ta ∼= S1, so that M is compact as well. Thus, the local diffeomorphismı : M → Ca ⊂ C must be a covering, and, in particular, Ca = C.

Thus, we have to consider those a ∈ g for which Ta∼= S1, Ca = C, and such that the action of

Ta on the universal cover of C is free.By a thorough investigation of the elements a ∈ g with the above properties, one finds that the

stabilizer S ⊂ G of a is a maximal compact subgroup such that Ta ⊂ S is the connected componentof the center. Since S also acts transitively on C, i.e. C = S/K for some compact subgroup K ⊂ S,it follows that M is a cover of Ta\S/K = S/(Ta · K), and finally, one shows that S/(Ta · K) ishermitian symmetric.

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