Remarks on Grassmannian Symmetric Spaces
aa r X i v : . [ m a t h . DG ] J a n REMARKS ON GRASSMANNIAN SYMMETRIC SPACES
LENKA ZALABOV ´A AND VOJTˇECH ˇZ ´ADN´IK
Abstract.
The classical concept of affine locally symmetric spaces allows ageneralization for various geometric structures on a smooth manifold. Weremind the notion of symmetry for parabolic geometries and we summarizethe known facts for | | –graded parabolic geometries and for almost Grassman-nian structures, in particular. As an application of two general constructionswith parabolic geometries, we present an example of non–flat Grassmanniansymmetric space. Next we observe there is a distinguished torsion–free affineconnection preserving the Grassmannian structure so that, with respect to thisconnection, the Grassmannian symmetric space is an affine symmetric spacein the classical sense. Introduction
Affine (locally) symmetric spaces present a very classical topic in differentialgeometry, see e.g. to [9, chapter XI] for all details. In particular, a smooth manifold M with an affine connection is called affine locally symmetric space if for eachpoint x ∈ M there is a local symmetry centered at x , i.e. a locally defined affinetransformation s x such that x is an isolated fixed point of s x and T x s x = − id. Thenotion of local symmetry is easily modified for various geometric structures andthere are attempts to understand these generalizations. For instance, there is a lotknown on the so–called projectively symmetric spaces, see e.g. [10] and referencestherein. In the framework of parabolic geometries, the general definition of localsymmetry fits nicely especially for | | –graded parabolic geometries which includesprojective, conformal, and almost Grassmannian geometries as particular examples.We open the article with a review of basic ideas and concepts for parabolic ge-ometries, including the notion of Weyl structures and normal coordinates. Then weintroduce local symmetries for general parabolic geometries and provide their char-acterization in homogeneous model, Proposition 2.5. There is a couple of generalresults for symmetric | | –graded parabolic geometries [12, 14] which we recover insection 3 in a slightly improved way. In particular, for a local symmetry centeredat x , Theorem 3.2 provides an existence of a torsion–free Weyl connection, locallydefined in a neighborhood of x , which is invariant with respect to the symmetryand whose Rho–tensor vanishes at x . Then we focus on Grassmannian (locally)symmetric spaces, i.e. almost Grassmannian structures allowing a (local) symme-try at each point. It turns out that the model Grassmannian structure is alwayssymmetric and a non–flat almost Grassmannian structure of type ( p, q ) may be lo-cally symmetric only if p or q is 2. In the latter case, the possible local symmetriesat a point are heavily restricted, see 3.4.The rest of the paper is devoted to the description of an example of non–flatGrassmannian locally symmetric space of type (2 , q ). This appears as the space of Mathematics Subject Classification.
Key words and phrases. parabolic geometries, Weyl structures, almost Grassmannian struc-tures, symmetric spaces.First author supported at different times by the Eduard ˇCech Center, project nr. LC505, andthe ESI Junior Fellows program; second author supported by the grant nr. 201/06/P379 of theGrant Agency of Czech Republic. chains of the homogeneous model of a parabolic contact geometry, Theorem 4.3.(The notion of chains here generalizes the Chern–Moser chains on CR manifolds ofhypersurface type.) The example comes as an application of some general construc-tions from [8] and [2], dealing with the parabolic geometry associated to the pathgeometry of chains. The necessary background for a comfortable understanding ofthe result is presented in 4.1 and 4.2. In addition, the constructed space is globallysymmetric and there is a torsion–free affine connection preserving the Grassman-nian structure which is invariant with respect to some distinguished symmetries,Theorem 4.4. Hence, we end up with an affine symmetric space with a compatibleGrassmannian structure, Corollary 4.4.
Acknowledgements.
We would like to mention the discussions with Andreas ˇCap,Boris Doubrov, and Jan Slov´ak, as well as the remarks by the anonymous referee,which were very helpful during the work on this paper.2.
Parabolic geometries, Weyl structures, and symmetries
In this section, we remind definitions and basic facts on Cartan geometries, Weylstructures and symmetries for parabolic geometries. We primarily refer to [3, 5, 6]for more intimate and comprehensive introduction to parabolic geometries, thesubsection dealing with symmetries is based on [12].2.1.
Definitions.
Let G be a Lie group, P ⊂ G its Lie subgroup, and p ⊂ g the corresponding Lie algebras. A Cartan geometry of type (
G, P ) on a smoothmanifold M is a couple ( G →
M, ω ) consisting of a principal P –bundle G → M together with a one–form ω ∈ Ω ( G , g ), which is P –equivariant, reproduces thefundamental vector fields and induces a linear isomorphism T u G ∼ = g for each u ∈ G .The one–form ω is called the Cartan connection . The curvature of Cartan geometryis defined as K := dω + [ ω, ω ], which is a two–form on G with values in g . Easily,the P –bundle G → G/P with the (left) Maurer–Cartan form µ ∈ Ω ( G, g ) forma Cartan geometry of type ( G, P ) with vanishing curvature, which we call the homogeneous model . Parabolic geometry is a Cartan geometry (
G →
M, ω ) of type (
G, P ), where G isa semisimple Lie group and P its parabolic subgroup. The Lie algebra g of the Liegroup G is equipped (up to the choice of Levi factor g in p ) with the grading of theform g = g − k ⊕· · ·⊕ g ⊕· · ·⊕ g k such that the Lie algebra p of P is p = g ⊕· · ·⊕ g k .Suppose the grading of g is fixed and further denote g − := g − k ⊕ · · · ⊕ g − and p + := g ⊕ · · · ⊕ g k . Parabolic geometry corresponding to the grading of length k is called | k | –graded . By G we denote the subgroup in P , with the Lie algebra g , consisting of all elements in P whose adjoint action preserves the grading of g .Next, defining P + := exp p + , we get P/P + = G and P = G ⋊ P + .The grading of g induces a P –invariant filtration g = g − k ⊃ g − k +1 ⊃ · · · ⊃ g k = g k , where g i := g i ⊕· · ·⊕ g k . This gives rise to a filtration of the tangent bundle T M as follows. The Cartan connection ω provides an identification T M ∼ = G × P ( g / p )where the action of P on g / p is induced by the adjoint representation. Hence each P –invariant subspace g − i / p ⊂ g / p defines the subbundle T − i M := G × P ( g − i / p ) in T M , so we obtain the filtration
T M = T − k M ⊃ · · · ⊃ T − M . Alternatively, thefiltration is described using the adjoint tractor bundle which is the natural bundle A M := G× P g corresponding to the (restriction of) adjoint representation of G on g .The filtration of g induces a filtration A M = A − k M ⊃ · · · ⊃ A M ⊃ · · · ⊃ A k M sothat T − i M ∼ = A − i M/ A M , in particular, T M ∼ = A M/ A M . Next by gr( T M ) wedenote the associated graded bundle gr(
T M ) = gr − k ( T M ) ⊕· · ·⊕ gr − ( T M ), wheregr − i ( T M ) := T − i M/T − i +1 M is the associated bundle to G with the standard fiber g − i / g − i +1 which is isomorphic to g − i as a G –module. Since P + ⊂ P acts freely on EMARKS ON GRASSMANNIAN SYMMETRIC SPACES 3 G , the quotient G /P + =: G is a principal bundle over M with the structure group P/P + = G . Hence gr( T M ) ∼ = G × G g − and the Lie bracket on g − induces analgebraic bracket on gr( T M ).By definition, the curvature K ∈ Ω ( G , g ) of a parabolic geometry is strictlyhorizontal and P –equivariant, hence it is fully described by a two–form on M with values in G × P g = A M , which is denoted by κ . Note that by the samesymbol we also denote the corresponding frame form, which is a P –equivariantmap G → ∧ ( g / p ) ∗ ⊗ g , the so–called curvature function . The Killing form on g provides an identification ( g / p ) ∗ ∼ = p + , hence the curvature function is viewedas having values in ∧ p + ⊗ g . The grading of g brings a grading to this spaceand parabolic geometry is called regular if the curvature function has values in thepart of positive homogeneity. The parabolic geometry is regular if and only if thealgebraic bracket on gr( T M ) above coincides with the
Levi bracket , which is thenatural bracket induced by the Lie bracket of vector fields. Next, the parabolicgeometry is called torsion–free if κ has values in ∧ p + ⊗ p ; note that torsion–freeparabolic geometry is automatically regular. Altogether, for a regular parabolicgeometry, there is an underlying structure on M consisting of a filtration of thetangent bundle (which is compatible with the Lie bracket of vector fields) and areduction of the structure group of gr( T M ) to the subgroup G .The correspondence between regular parabolic geometries of specified type andthe underlying structures can be made bijective (up to isomorphism) provided oneimpose some normalization condition: The parabolic geometry is called normal if ∂ ∗ ◦ κ = 0, where ∂ ∗ : ∧ p + ⊗ g → p + ⊗ g is the differential in the standardcomplex computing the homology H ∗ ( p + , g ) of p + with coefficients in g . Dealingwith regular normal parabolic geometries, there is the notion of harmonic curvature κ H , which is the composition of κ with the natural projection ker( ∂ ∗ ) → H ( p + , g ).By definition, κ H is a section of G × P H ( p + , g ) and, since P + acts trivially on H ∗ ( p + , g ), it can be interpreted in terms of the underlying structure. The mainissue is that the harmonic curvature κ H is much simpler object than the curvature κ , however, still involving the whole information about κ . Note that, as a G –module, each H j ( p + , g ) is isomorphic to ker (cid:3) ⊂ ker ∂ ∗ ⊂ ∧ j p + ⊗ g , the kernelof the Kostant Laplacian. As a consequence of [4, Corollary 4.10], which is anapplication of generalized Bianchi identity, we conclude: Lemma.
Let κ and κ H be the Cartan curvature and the harmonic curvature ofa regular normal parabolic geometry of type ( G, P ) . Then the lowest non–zerohomogeneous component of κ has values in ker (cid:3) ⊂ ∧ p + ⊗ g , i.e. it coincides withthe corresponding homogeneous component of κ H . In particular, κ H = 0 if andonly if κ = 0 . If κ = 0, the parabolic geometry is called flat (or locally flat ). Flat parabolicgeometry of type ( G, P ) is locally isomorphic to the homogeneous model ( G → G/P, µ ).2.2.
Examples.
Here we focus on two classes of parabolic geometries which areoften mentioned in the sequel:(1) An important family of examples is formed by | | –graded parabolic geome-tries. Any | | –graded parabolic geometry is trivially regular and the main featureof any such geometry is that the tangent bundle T M has not got any nontrivialnatural filtration. Hence (up to one exception) the underlying structure on M isjust a classical first order G –structure. All the section 3 deals with | | –gradedparabolic geometries, with almost Grassmannian structures in particular.(2) Another interesting examples are the parabolic contact geometries , whichare | | –graded parabolic geometries with underlying contact structure. Parabolic LENKA ZALABOV´A AND VOJTˇECH ˇZ´ADN´IK contact geometry corresponds to a contact grading of a simple Lie algebra g , whichis a grading g = g − ⊕ g − ⊕ g ⊕ g ⊕ g such that g − is one dimensional andthe Lie bracket [ , ] : g − × g − → g − is non–degenerate. The filtration of T M looks like
T M = T − M ⊃ T − M so that D := T − M is the contact distribution.For regular parabolic contact geometries, the Levi bracket L : D × D →
T M/ D is non–degenerate and the reduction of gr( T M ) = (
T M/ D ) ⊕ D to the structuregroup G corresponds to an additional structure on D .The best known examples of parabolic contact geometries are non–degeneratepartially integrable almost CR structures of hypersurface type where the addi-tional structure on D is an almost complex structure. Another examples are theLagrangean contact structures which are introduced in 4.3 in some detail.2.3. Weyl structures and connections.
Let (
G →
M, ω ) be a parabolic geom-etry of type (
G, P ), let G = G /P + be the underlying G –bundle as in 2.1, and let π : G → G be the canonical projection. A Weyl structure of the parabolic geom-etry is a global smooth G –equivariant section σ : G → G of the projection π . Inparticular, any Weyl structure provides a reduction of the principal bundle G → M to the subgroup G ⊂ P . For arbitrary parabolic geometry, Weyl structures alwaysexist and any two Weyl structures σ and ˆ σ differ by a G –equivariant mappingΥ : G → p + so that ˆ σ ( u ) = σ ( u ) · exp Υ( u ), for all u ∈ G . Since Υ is the frameform of a one–form on M , all Weyl structures form an affine space modelled overΩ ( M ) and the relation above is simply written as ˆ σ = σ + Υ.Denote by ω i the g i –component of the Cartan connection ω ∈ Ω ( G , g ). Thechoice of the Weyl structure σ defines the collection of G –equivariant one–forms σ ∗ ω i ∈ Ω ( G , g i ). The one–form σ ∗ ω reproduces the fundamental vector fieldsof the principal action of G on G , hence it defines a principal connection on G which we call the Weyl connection of the Weyl structure σ . The Weyl connectioninduces connections on all bundles associated to G and these are often called bythe same name. For any i = 0, the one–form σ ∗ ω i is strictly horizontal, hence itdescends to a one–form on M with values in A i M := A i M/ A i +1 M . In particular,the whole negative part σ ∗ ω − = σ ∗ ω − k ⊕ · · · ⊕ σ ∗ ω − , which is called the solderingform , provides an identification of the tangent bundle T M with the associatedgraded tangent bundle gr(
T M ) ∼ = A − k M ⊕ · · · ⊕ A − M . The positive part σ ∗ ω + = σ ∗ ω ⊕ · · · ⊕ σ ∗ ω k is called the Rho–tensor and denoted as P . The Rho–tensor isused to compare the Cartan connection ω on G and the principal connection on G extending the Weyl connection σ ∗ ω from the image of σ : G → G . By definition, P is a one–form on M with values in A M ⊕ · · · ⊕ A k M and since this bundle isidentified with T ∗ M , the Rho–tensor can be viewed as a section of T ∗ M ⊗ T ∗ M .Among general Weyl structures, there are various specific subclasses. We focuson the so–called normal Weyl structures which play some role in the sequel. NormalWeyl structures are related to the notion of normal coordinates as follows. Givena parabolic geometry ( p : G →
M, ω ) of type (
G, P ) and a fixed element u ∈ G , the normal coordinates at x = p ( u ) is the local diffeomorphism Φ u from a neighbor-hood U of 0 ∈ g − to a neighborhood of x ∈ M , defined by X p (Fl ω − ( X )1 ( u )).(By ω − ( X ) we denote the constant vector field on G corresponding to X .) Now,over the image Φ u ( U ) ⊂ M , there is a unique G –equivariant section σ u : G → G such that Fl ω − ( U )1 ( u ) ⊂ σ u ( G ), which we call the normal Weyl structure at x .Although the normal Weyl structure is indexed by u ∈ G , it obviously dependsonly on the orbit of u ∈ p − ( x ) by the action of G . If ∇ and P is the corre-sponding affine connection and Rho–tensor, respectively, then the normal Weylstructure σ u is characterized by the property that for all k ∈ N the symmetrization EMARKS ON GRASSMANNIAN SYMMETRIC SPACES 5 of ( ∇ ξ k . . . ∇ ξ P )( ξ ) over all ξ i ∈ T M vanishes at x = p ( u ). Hence, in particular, P ( x ) = 0, cf. [5, Theorem 3.16].2.4. Automorphisms and symmetries. An automorphism of Cartan geometry( G →
M, ω ) of type (
G, P ) is a principal bundle automorphism ϕ : G → G such that ϕ ∗ ω = ω . It is well known that all automorphisms of (a connected component of)the homogeneous model ( G → G/P, µ ) are just the left multiplications by elementsof G . Any g ∈ G induces a base map ℓ g : G/P → G/P and it turns out that twoelements of G have got the same base map if and only if they differ by an elementfrom the kernel K of the pair ( G, P ), which is the maximal normal subgroup of G contained in P . (If K is trivial then G acts effectively on G/P .) Moreover, thesame characterization holds also for general Cartan geometries, [11, chapters 4 and5]. In the cases of parabolic geometries, the kernel K is always discrete and veryoften finite if not trivial. An automorphism ϕ : G → G of parabolic geometry isthen uniquely determined by its base map ϕ : M → M up to a smooth equivariantfunction G → K which has to be constant over connected components of M . Definition.
Let (
G →
M, ω ) be a regular | k | –graded parabolic geometry, let T M = T − k M ⊃ · · · ⊃ T − M be the corresponding filtration of the tangent bundle, andlet x ∈ M be a point. A local symmetry of the parabolic geometry centered at x isa locally defined diffeomorphism s x of a neighborhood of x such that:(i) s x ( x ) = x ,(ii) T x s x | T − x M = − id T − x M ,(iii) s x is covered by an automorphism of the parabolic geometry.If the local symmetry can be extended to a global symmetry on M , we just speakabout symmetry . The parabolic geometry is called ( locally ) symmetric if there is a(local) symmetry at each point x ∈ M .Note that for | | –graded parabolic geometries the restriction in the condition (ii)above is actually superfluous since T − M = T M . Hence it can be shown that s x isinvolutive and x is an isolated fixed point. In this view, the definition above reflectsthe classical notion of affine locally symmetric spaces. The main difference to theclassical issues is that parabolic geometries are not structures of first order, hence,in particular, the conditions above do not determine the symmetry uniquely.Note also that the condition (ii) cannot be extended to the whole T M in gen-eral: Any symmetry is by definition covered by an automorphism of the para-bolic geometry, hence it has to preserve the underlying structure. For instance,consider a parabolic contact geometry introduced in example 2.2(2). In particu-lar, the underlying structure comprise of the contact distribution
D ⊂
T M andthe non–degenerate Levi bracket L : D × D →
T M/ D . If there was a map s satisfying (i), (iii), and T x s = − id T x M , then for any ξ, η ∈ D x it would hold s ( L ( ξ, η )) = L ( s ( ξ ) , s ( η )) = L ( ξ, η ) and, simultaneously, s ( L ( ξ, η )) = −L ( ξ, η ),which would contradict the non–degeneracy of L .2.5. Symmetries of homogeneous models.
Let ( G → G/P, µ ) be the homo-geneous model of a parabolic geometry of type (
G, P ) and let
G/P be connected.As we mentioned in the beginning of 2.4, all automorphisms of the homogeneousmodel are just the left multiplications by elements of G . Next, an analog of theLiouville theorem states that any local automorphism can be uniquely extendedto a global one. Hence if the homogeneous model is locally symmetric then it issymmetric. By the transitivity of the action of G on G/P and the above character-ization of the automorphisms, in order to decide whether the homogeneous modelis symmetric, it suffices to find a symmetry at the origin. Due to the identification
LENKA ZALABOV´A AND VOJTˇECH ˇZ´ADN´IK T ( G/P ) ∼ = G × P ( g / p ) as in 2.1, the previous task is equivalent to find an elementin P which acts as − id on g − / p ⊂ g / p . Since P = G ⋊ P + and P + acts triv-ially on g − / p , one is actually looking for an element of G acting as − id on g − .Altogether, we have got the following general statement: Proposition.
All symmetries of the homogeneous model ( G → G/P, µ ) of a para-bolic geometry of type ( G, P ) centered at any point are parametrized by the elements g exp Z ∈ P , where Z ∈ p + is arbitrary and g ∈ G such that Ad g | g − = − id | g − .In particular, if there is one symmetry at a point then there is an infinite amountof them. It is usually a simple exercise to find all elements from G with the propertyas above. Note that different choices of the pair of Lie groups ( G, P ) with theLie algebras p ⊂ g may lead to different amount of such elements. This actuallycorresponds to the cardinality of the kernel K , as defined in 2.4.3. | | –graded and Grassmannian locally symmetric spaces Firstly we collect the facts on symmetries which hold for general | | –gradedparabolic geometries. Then we focus on Grassmannian locally symmetric spacesand provide a discussion which is specific in that case. In any case, the parabolicgeometry in question is | | –graded, so the tangent map to a possible symmetry at x ∈ M acts as − id on all of T x M . Hence the following fact is obvious and oftenused below:3.1. Lemma.
For a | | –graded parabolic geometry on M , tensor field of odd degreewhich is invariant with respect to a symmetry at x ∈ M vanishes at x . General restrictions.
Following [5, section 4], we start with a bit of notationwe use below. Let (
G →
M, ω ) be a normal | | –graded parabolic geometry, let κ be the Cartan curvature, and let κ H be its harmonic curvature. Let σ : G → G be a Weyl structure, and let τ i := σ ∗ ω i be the corresponding Weyl forms as in 2.3.Let us consider the curvature dτ + [ τ, τ ] = σ ∗ κ ∈ Ω ( G , g ) and its decomposition T + W + Y according to the values in g − ⊕ g ⊕ g = g . As before, T, W , and Y is represented by a two–form on M with values in A − M ∼ = T M , A M ∼ =End ( T M ), and A M ∼ = T ∗ M , respectively. By definition, T = dτ − + [ τ − , τ ],hence it coincides with the torsion of the affine connection ∇ on M induced by theWeyl connection τ . By lemma 2.1, this further coincides with the homogeneouscomponent of degree one of the harmonic curvature κ H , hence it is independent ofthe choice of Weyl structure. Similarly, W = dτ + [ τ , τ ] + [ τ − , τ ], where thefirst two summands represent just the curvature R of ∇ . Since τ = P , the lastsummand is rewritten as ∂ P , where ( ∂ P )( ξ, η ) = { ξ, P ( η ) } + { P ( ξ ) , η } , where { , } is the algebraic bracket on A M given by the Lie bracket in g . Altogether,(1) W = R + ∂ P and we call W the Weyl curvature . If T = 0 then W coincides by lemma 2.1 withthe homogeneous component of κ H of degree two and so it is invariant with respectto the change of Weyl structure.As an immediate application of lemma 3.1 we have got the following: Proposition.
If a | | –graded parabolic geometry is locally symmetric then it istorsion–free. In particular, any underlying Weyl connection is torsion–free. Now we are going to find some information on the curvature of symmetric | | –graded parabolic geometries. If ϕ is an automorphism of the parabolic geometrythen for arbitrary Weyl structure ˆ σ the pullback ϕ ∗ ˆ σ is again Weyl structure, hence ϕ ∗ ˆ σ = ˆ σ + Υ, for some uniquely given one–form Υ. If ϕ in addition covers some EMARKS ON GRASSMANNIAN SYMMETRIC SPACES 7 symmetry at x then one checks that the Weyl structure σ := ˆ σ + Υ satisfies ϕ ∗ σ = σ in the fiber over x . (Restricted to the fiber over x , the definition of σ doesnot depend on ˆ σ .) Next, let ¯ σ be the normal Weyl structure at x which is uniquelydetermined by σ over x . Since ϕ ∗ ¯ σ is again normal and by construction ϕ ∗ ¯ σ = ¯ σ over x , it has to coincide with ¯ σ on its domain. Altogether, [14, sections 9 and 10]: Lemma.
Let ϕ be an automorphism of | | –graded parabolic geometry which coversa local symmetry at a point x . Then (1) there is a Weyl structure which is invariant under ϕ over the point x , (2) there is a unique normal Weyl structure which is invariant under ϕ oversome neighborhood of x . Let ∇ be the affine connection corresponding to the normal Weyl structure ona neighborhood of x as above and let T, R, W , and P be its torsion, curvature,Weyl curvature, and Rho–tensor, respectively. By construction, the connection ∇ is invariant with respect to the local symmetry at x and by the previous Proposition, T = 0. Similarly, ∇ W is also invariant under the symmetry at x , however, it is atensor field of degree five which vanishes at x by lemma 3.1. Since ∇ is normal at x , P vanishes at x , hence from the equation (1) we conclude that also ∇ R = 0 at x : Theorem.
Suppose there is a local symmetry s x of a | | –graded parabolic geometrycentered at x . Then, on a neighborhood of x , there exists a torsion–free Weylconnection ∇ which is invariant under s x and whose Rho–tensor vanishes at x .Consequently, ∇ R vanishes at x . Almost Grassmannian structures.
The notion of almost Grassmannianstructure on a smooth manifold generalizes the geometry of Grassmannians. The
Grassmannian of type ( p, q ) is the space Gr( p, R p + q ) of p –dimensional linear sub-spaces in the real vector space R p + q . It is well known that, for any E ∈ Gr( p, R p + q ),the tangent space of Gr( p, R p + q ) in E is identified with E ∗ ⊗ ( R p + q /E ), the spaceof linear maps from E to the quotient. In particular, the dimension of Gr( p, R p + q )is pq . Note that Gr(1 , R q ) is the real projective space RP q , so we always as-sume p > p, R p + q )may be considered as the space of ( p − RP p + q − = P ( R p + q ). The Lie group ˆ G := P GL ( p + q, R ) acts transitively (andeffectively) on Gr( p, R p + q ) and the stabilizer of a fixed element is the parabolicsubgroup ˆ P as described below. The Grassmannian of type ( p, q ) is then the ho-mogeneous model of the parabolic geometry of type ( ˆ G, ˆ P ).The Lie algebra of ˆ G is ˆ g = sl ( p + q, R ) and let us consider its grading which isschematically described by the block decomposition (cid:18) ˆ g ˆ g ˆ g − ˆ g (cid:19) with blocks of sizes p and q along the diagonal. In particular, ˆ g − ∼ = R p ∗ ⊗ R q ,ˆ g ∼ = s ( gl ( p, R ) ⊕ gl ( q, R )), and ˆ g ∼ = R p ⊗ R q ∗ . The parabolic subgroup ˆ P ⊂ ˆ G isrepresented by block upper triangular matrices with the Lie algebra ˆ p = ˆ g ⊕ ˆ g ,the subgroup ˆ G corresponds then to the block diagonal matrices in ˆ P .An almost Grassmannian structure of type ( p, q ) on a smooth manifold M isdefined to be a | | –graded parabolic geometry of type ( ˆ G, ˆ P ) where the groups are asabove. The underlying structure on M is equivalent to the choice of auxiliary vectorbundles E → M and F → M of rank p and q , respectively, and an isomorphism E ∗ ⊗ F → T M . Note that a different choice of the Lie group to the Lie algebraˆ g = sl ( p + q, R ) gives rise to an additional structure on M . In particular, the usual LENKA ZALABOV´A AND VOJTˇECH ˇZ´ADN´IK choice for ˆ G to be SL ( p + q, R ) leads to a preferred trivialisation of ∧ p E ⊗ ∧ q F which is often supposed in the literature. In contrast to the previous choice, thisgroup has got a non–trivial center provided p + q is even.3.4. Grassmannian locally symmetric spaces.
When we speak about a
Grass-mannian (locally) symmetric space , we mean a smooth manifold with an almostGrassmannian structure which is (locally) symmetric in the sense of 2.4. For tech-nical reasons we always assume the almost Grassmannian structure is representedby a normal parabolic geometry of type ( ˆ G, ˆ P ), which is uniquely determined by theunderlying structure up to isomorphism. Note that by Proposition 3.2, any Grass-mannian locally symmetric space admits a torsion–free affine connection preservingthe structure, hence the almost Grassmannian structure is actually Grassmannian,i.e. it is integrable in the sense of G –structures.According to Proposition 2.5, it is an easy exercise to decide whether the ho-mogeneous model ˆ G/ ˆ P is symmetric or not. A direct calculation shows that, [14,section 7]: Proposition.
The homogeneous model of almost Grassmannian structures of type ( p, q ) is always symmetric. An explicit description of the harmonic curvature in individual cases yields thatif p > q >
Proposition.
Grassmannian locally symmetric space of type ( p > , q > is flat,i.e. locally isomorphic to the homogeneous model. Hence the only non–flat almost Grassmannian structures which can carry (local)symmetries are of type ( p, q ) where p or q is 2; this is always supposed hereafter.In all these cases, the harmonic curvature has two components which are mostly ofhomogeneity one and two. (Note that the extremal case p = q = 2 has a specificfeature, namely, there are two components of homogeneity two. Moreover, analmost Grassmannian structure of type (2 ,
2) is equivalent to a conformal pseudo–Riemannian structure of the split signature, [3, section 3.5].) Since the torsion partvanishes for any Grassmannian locally symmetric space, the remaining componentof homogeneous degree two corresponds to the Weyl curvature which is then theonly obstruction to the local flatness of the structure.From 3.2 we know that the existence of a local symmetry at a point x yieldssome restriction on the Weyl curvature at that point. This heavily forces thefreedom for another possible symmetries at x : Suppose there are two different localsymmetries at x which are covered by ϕ and ϕ . Let σ and σ be the Weylstructures associated to ϕ and ϕ by lemma 3.2 and let Υ be the one–form suchthat σ = σ + Υ. In this way, two different symmetries at x defines an element Υ x in T ∗ x M ∼ = E x ⊗ F ∗ x , which turns out to be non–zero. With this notation, it can beproved the following, [13, section 4.3]: Theorem.
Let M be a smooth manifold with an almost Grassmannian structure oftype (2 , q ) or ( p, . If there are two different local symmetries centered at a point x ∈ M and the corresponding covector Υ x constructed above has maximal rank thenthe Weyl curvature vanishes at x and, consequently, the Cartan curvature vanishesat x . (Note that the Theorem is originally formulated with respect to a one–formwhich was constructed in a different way. However, it is easy to check the twoone–forms coincide up to a non–zero multiple at x .) EMARKS ON GRASSMANNIAN SYMMETRIC SPACES 9 The example
In this section we present an example of non–flat homogeneous Grassmanniansymmetric space. By the previous results, this has to be necessarily either of type(2 , q ) or of type ( p, Path geometry of chains.
Let (
G →
M, ω ) be a contact parabolic geometryof type (
G, P ) and let g be the corresponding Lie algebra with the contact gradingas in example 2.2(2). The 1–dimensional subspace g − ⊂ g − gives rise to a familyof distinguished curves on M which are called the chains and which play a crucialrole in the sequel. More specifically, chains are defined as projections of flow lines ofconstant vector fields on G corresponding to non–zero elements of g − . Equivalently,using the notion of development of curves, chains are the curves which develop tomodel chains in the homogeneous model G/P . The latter curves passing throughthe origin are the curves of type t b exp( tX ) P , for b ∈ P and X ∈ g − . Asnon–parametrized curves, chains are uniquely determined by a tangent direction ina point, [7, section 4]. By definition, chains are defined only for directions whichare transverse to the contact distribution D ⊂
T M . In classical terms, the familyof chains defines a path geometry on M restricted, however, only to the directionstransverse to D .A path geometry on M is equivalent to a decomposition Ξ = E ⊕ V of thetautological subbundle Ξ ⊂ T P T M where V is the vertical subbundle of the obviousprojection P T M → M and E is a fixed transversal line subbundle. (Given such adecomposition, the paths in the family are the projections of integral submanifoldsof the distribution E .) Lie bracket of vector fields behave specifically with respect tothe decomposition above and it turns out this structure on P T M can be described asa parabolic geometry of type ( ˜ G, ˜ P ), where ˜ G = P GL ( m +1 , R ), m = dim M , and ˜ P is the parabolic subgroup as follows. Let us consider the grading of ˜ g = sl ( m + 1 , R )which is schematically described by the block decomposition with blocks of sizes 1,1, and m − ˜ g ˜ g E ˜ g ˜ g E − ˜ g ˜ g V ˜ g − ˜ g V − ˜ g . Then ˜ p := ˜ g ⊕ ˜ g ⊕ ˜ g is a parabolic subalgebra of ˜ g and ˜ P ⊂ ˜ G is the subgrouprepresented by block upper triangular matrices so that its Lie algebra is ˜ p . Asusual, the parabolic geometry associated to the path geometry on M is uniquelydetermined (up to isomorphism) provided we consider it is regular and normal inthe sense of 2.1.Back to the initial setting, given a contact manifold M with a parabolic contactgeometry of type ( G, P ), the path geometry of chains gives rise to a parabolicgeometry of type ( ˜ G, ˜ P ) restricted to the open subset ˜ M ⊂ P T M consisting of alllines which are transverse to the contact distribution
D ⊂
T M . Let Q ⊂ P bethe subgroup which stabilizes the subspace g − ⊂ g − under the action of P on g − induced from the adjoint action on g ; the Lie algebra of Q is evidently q = g ⊕ g .By [8, lemma 2.2], the space ˜ M of all lines in T M transverse to D is identified withthe orbit space G /Q .Altogether, for a parabolic contact structure on M given by a regular and normalparabolic geometry ( G →
M, ω ) of type (
G, P ), let ( ˜
G → ˜ M , ˜ ω ) be the regularnormal parabolic geometry of type ( ˜ G, ˜ P ) corresponding to the path geometry of chains. Due to the identification ˜ M ∼ = G /Q , the couple ( G → ˜ M , ω ) forms aCartan (but not parabolic) geometry of type (
G, Q ). In some cases, the two Cartangeometries over ˜ M can be directly related by a pair of maps ( i : Q → ˜ P , α : g → ˜ g )so that ˜ G ∼ = G × Q ˜ P and j ∗ ˜ ω = α ◦ ω , where j is the canonical inclusion G ֒ → G × Q ˜ P .The two maps ( i, α ) has to be compatible in some strong sense by the equivariancyof j and the fact that both ω and ˜ ω are Cartan connections, [8, Proposition 3.1].On the other hand, any pair of maps ( i, α ) which are compatible in the above sensegives rise to a functor from Cartan geometries of type ( G, Q ) to Cartan geometriesof type ( ˜ G, ˜ P ) and there is a perfect control over the natural equivalence of functorsassociated to different pairs. For what follows, we need to understand the effect ofsuch construction on the curvature of the induced Cartan geometry. In particular,[8, Proposition 3.3] shows that: Lemma.
Let a flat Cartan geometry of type ( G, Q ) be given. Then the Cartangeometry of type ( ˜ G, ˜ P ) induced by the pair ( i, α ) is flat if and only if α is a homo-morphism of Lie algebras. Correspondence spaces and twistor spaces.
Below we enjoy an applica-tion of another general construction relating parabolic geometries of different types,namely, the construction of correspondence spaces and twistor spaces in the senseof [2] or [3], to which we refer for all details.Let ˜ G be a semisimple Lie group and ˜ P ⊂ ˜ P ⊂ ˜ G parabolic subgroups. If aparabolic geometry ( ˜ G → ˜ N , ˜ ω ) of type ( ˜ G, ˜ P ) on a smooth manifold ˜ N is given,then the correspondence space of ˜ N corresponding to the subgroup ˜ P ⊂ ˜ P isdefined as the orbit space C ˜ N := ˜ G / ˜ P . The couple ( ˜ G → C ˜ N , ˜ ω ) forms a parabolicgeometry of type ( ˜ G, ˜ P ). Let V ⊂ T C ˜ N be the vertical subbundle of the naturalprojection C ˜ N → ˜ N . Then easily, i ξ ˜ κ = 0 for any ξ ∈ V , where ˜ κ is the Cartancurvature of ˜ ω . Note that V corresponds to the ˜ P –invariant subspace ˜ p / ˜ p ⊂ ˜ g / ˜ p under the identification T C ˜ N ∼ = ˜ G × ˜ P (˜ g / ˜ p ).Conversely, given a parabolic geometry ( ˜ G → ˜ M , ˜ ω ) of type ( ˜ G, ˜ P ), let ˜ κ be itsCartan curvature, and let V ⊂ T ˜ M be the distribution corresponding to ˜ p / ˜ p ⊂ ˜ g / ˜ p . Then, locally, ˜ M is a correspondence space of a parabolic geometry of type( ˜ G, ˜ P ) if and only if i ξ ˜ κ = 0 for all ξ ∈ V , [3, Theorem 3.3]. Note that this conditionin particular implies the distribution V is integrable and the local leaf space of thecorresponding foliation is called the twistor space . The parabolic geometry of type( ˜ G, ˜ P ) is locally formed over the corresponding twistor space.Note that the present considerations does not restrict only to parabolic geome-tries, as we actually partially observed in the previous subsection. Still, for para-bolic geometries the constructions above are always compatible with the normalitycondition. Concerning the regularity, this is not true in general, but an efficientcontrol of this condition is usually very easy. Dealing with a regular normal para-bolic geometry of type ( ˜ G, ˜ P ), let ˜ κ H be the harmonic curvature and let V ⊂ T ˜ M be as above. Then there is the following useful simplification of the previous char-acterization of correspondence spaces, [3, Proposition 3.3]: If i ξ ˜ κ H = 0 for all ξ ∈ V then i ξ ˜ κ = 0 for all ξ ∈ V . Example.
Let ( ˜
G → P
T M, ˜ ω ) be the Cartan geometry associated to a path geom-etry on M , i.e. a parabolic geometry of type ( ˜ G, ˜ P ) with the notation as in 4.1. Letˆ P ⊂ ˜ G be the subgroup (containing ˜ P and) consisting of block upper triangularmatrices with Lie algebra ˆ p = ˜ g E − ⊕ ˜ p according to the description above. Note thatthe underlying structure of a parabolic geometry of type ( ˜ G, ˆ P ) is just the Grass-mannian structure of type (2 , q ), where q = dim M −
1, cf. the definition in 3.3.The distribution in T P T M corresponding to the linear subspace ˆ p / ˜ p ⊂ ˜ g / ˜ p is just EMARKS ON GRASSMANNIAN SYMMETRIC SPACES 11 the line subbundle E determined by the path geometry on M , in particular this isalways involutive. Hence the corresponding local twistor space ˜ N coincides locallywith the space of paths of the path geometry. From the above characterization ofcorrespondence spaces and the explicit description of the irreducible components ofthe harmonic curvature ˜ κ H of the Cartan connection ˜ ω , it follows that [3, example3.4]: Lemma.
Let ( ˜
G → P
T M, ˜ ω ) be a Cartan geometry of type ( ˜ G, ˜ P ) and let ˜ N bethe local twistor space as above. Then the Cartan geometry on P T M descends toa Grassmannian structure on ˜ N if and only if ˜ ω is torsion–free. Applications.
Now, the promised example of a Grassmannian symmetricspace appears as an application of the general principles described in previousparagraphs. We are going to start with the model Lagrangean contact structure,however the analogous ideas work for another parabolic contact structures as well.This is discussed in remark 4.5(4) where we also highlight the differences.A
Lagrangean contact structure on a smooth manifold M of odd dimension m =2 n + 1 consists of a contact distribution D ⊂
T M with a decomposition D = L ⊕ R such that the subbundles are isotropic with respect to the Levi bracket L : D ×D →
T M/ D . Lagrangean contact structure is an instance of parabolic contact structurecorresponding to the contact grading of simple Lie algebra g = sl ( n + 2 , R ), whichis schematically indicated by the following block decomposition with blocks of sizes1, n , and 1 along the diagonal: g g L g g L − g g R g − g R − g . As in general, the subspace g − defines the contact distribution, however now it issplit as g − = g L − ⊕ g R − such that this splitting is invariant under the adjoint actionof g and the subspaces g L − and g R − are isotropic with respect to the restrictedLie bracket [ , ] : g − × g − → g − . Let G = P GL ( n + 2 , R ) be the Lie groupwith Lie algebra g and let P ⊂ G be the subgroup represented by block uppertriangular matrices with the Lie algebra p = g ⊕ g ⊕ g . The homogeneousspace G/P is identified with P T ∗ RP n +1 , the projectivized cotangent bundle of realprojective space of dimension n +1, and the model Lagrangean contact structure on P T ∗ RP n +1 is induced from the flat projective structure on RP n +1 . Note that thiscorrespondence is just another instance of the correspondence space constructionsfrom 4.2, see [2, section 4.1].Now, put M = G/P and follow the construction from 4.1:(1) The subset ˜ M = P T M , consisting of all lines in
T M which are transverseto the contact distribution D , is identified with the homogeneous space G/Q .(2) The flat parabolic geometry ( G → G/P, µ ) of type (
G, P ), for µ being theMaurer–Cartan form on G , defines the flat Cartan geometry ( G → G/Q, µ ) of type(
G, Q ).(3) The latter induces the parabolic geometry ( G × Q ˜ P → G/Q, ˜ ω ) of type ( ˜ G, ˜ P )via the pair of maps ( i, α ) which are explicitly given in [8, section 3.5]. Note that α : g → ˜ g is not a homomorphism of Lie algebras, hence by lemma 4.1 the inducedCartan geometry is not flat. By [8, section 3.6], this is the unique regular normalparabolic geometry associated to the path geometry of chains: Lemma.
Let ( G → G/Q, µ ) be the flat Cartan geometry of type ( G, Q ) and let ( i, α ) be the pair of maps as in step (3) above. Then the induced parabolic geometry ( G × Q ˜ P → G/Q, ˜ ω ) is a non–flat torsion–free (and hence regular) normal parabolicgeometry of type ( ˜ G, ˜ P ) . (4) Finally, let ˜ N be the space of all chains on M = G/P , understood as non–parametrized curves as above. By definition, this is a locally defined leaf space ofthe foliation of ˜ M corresponding to the distribution E as in 4.2. In this model case,˜ N is a homogeneous space and it turns out to be a Grassmannian symmetric spacewhich is not flat, i.e. not locally isomorphic to the homogeneous model ˜ G/ ˆ P : Theorem.
Let M = G/P be the model Lagrangean contact structure. Then thespace ˜ N of all chains in M is a non–flat homogeneous Grassmannian symmetricspace of type (2 , q ) , where q = dim M − .Proof. Almost everything follows immediately from the previous profound prepa-ration:Lemmas 4.3 and 4.2 yield that ˜ N is endowed with a Grassmannian structureand the fact that the induced Cartan geometry of type ( ˜ G, ˜ P ) on ˜ M = G/Q has anon–trivial curvature implies the curvature of the corresponding Cartan geometryon the twistor space ˜ N is non–trivial as well. Since ˜ M = P T M is a homogeneousspace and any chain is uniquely determined by an element of ˜ M , the group G acts transitively on the space ˜ N of all chains. Let H ⊂ G be the stabilizer of thechain exp tX · P , X ∈ g − , passing through the origin in M = G/P . An easy directcomputation shows that H is the subgroup consisting of block matrices in G so thatits Lie algebra is h = g − ⊕ g ⊕ g , i.e. H = exp g − ⋉ Q . Altogether, ˜ N ∼ = G/H and consequently T ˜ N ∼ = G × H ( g / h ).By the very construction, elements of G act as automorphisms of the inducedCartan geometry on ˜ M and since the quotient ˆ P / ˜ P is obviously connected, thesedescend to automorphisms of the Grassmannian structure on ˜ N by [2, remark 2.4].In order to show there is a symmetry at any point of ˜ N , it suffices to find an elementin H which acts as − id on g / h . However, this is rather easy task and after a whileof calculation one shows that the block matrix − I n
00 0 − represents the unique element with this property. (cid:3) Remark.
In the proof above we have constructed a global symmetry of the Grass-mannian structure at the origin of ˜ N = G/H , which leads to a distinguished sym-metry at each point. Any such symmetry is represented by an element of G andit will be called the G –symmetry . Any G –symmetry primarily defines an auto-morphism of the Lagrangean contact structure on M = G/P and, easily, this is asymmetry on M in the sense of 2.4. Since the parabolic contact structure on M is flat, there is a lot of symmetries at any point, but only one of them induces asymmetry on ˜ N . On the other hand, apart from the G –symmetry, there may beanother local symmetries at any point of ˜ N . However, according to Theorem 3.4, allthe possible symmetries may differ from the G –symmetry in a very restricted sense.More specifically, by the homogeneity of the induced Grassmannian structure on˜ N , the corresponding Weyl curvature is nowhere vanishing, hence the covector Υ x from Theorem 3.4 measuring the difference of two symmetries at x must be of rankone.4.4. Invariant connection.
Let us conclude by a discussion on an affine connec-tion on ˜ N which preserves the Grassmannian structure and which is invariant withrespect to some symmetries. Note that the following statement can be seen asan instance of [1, Theorem 1] which deals with invariant connections on reductivehomogeneous spaces with a compatible additional structure of general | | –graded EMARKS ON GRASSMANNIAN SYMMETRIC SPACES 13 parabolic geometry. Of course, in our specific setting we can approach the resultin a more direct way.
Theorem.
Let ˜ N = G/H be the space of chains of model Lagrangean contactstructure on M = G/P . Then there is a G –invariant torsion–free affine connection ˜ ∇ on ˜ N preserving the Grassmannian structure.Proof. As we know from 4.2, the construction of twistor space and the correspond-ing Cartan geometry is very local in nature. However, in our model case, thereis the global surjective submersion p : ˜ M = G/Q → G/H = ˜ N to the twistorspace. The principal ˜ P –bundle π : G × Q ˜ P → ˜ M is defined according to the Liegroup homomorphism i : Q → ˜ P , which we referred to in step (3) in 4.3. Thehomomorphism i can be extended to a homomorphism ˆ i : H → ˆ P so that ˆ i ′ = α | h ,the total space of the principal bundle G × Q ˜ P → ˜ M is identified with G × H ˆ P ,and the composition p ◦ π : G × H ˆ P → ˜ N is a principal ˆ P –bundle. The proper-ties of the Cartan connection ˜ ω as above, namely the torsion freeness, yield thecouple ( G × H ˆ P → ˜ N , ˜ ω ) is a parabolic geometry of type ( ˜ G, ˆ P ) and ˜ M is thecorrespondence space for ˜ P ⊂ ˆ P .Now, let ˆ G be the Lie subgroup of ˜ G as in 3.3. Namely, ˆ G is represented byblock diagonal matrices with the Lie algebra ˆ g = ˜ g E − ⊕ ˜ g ⊕ ˜ g E , i.e. the reductivepart of ˆ p . Since ˆ i : H → ˆ P is a homomorphism of Lie groups, ˆ i ′ = α | h , and α ( h ) ⊂ ˆ g , it follows that ˆ i ( H ) ⊂ ˆ G . This gives rise to the principal ˆ G –bundle G × H ˆ G → ˜ N , which is a distinguished reduction of G × H ˆ P → ˜ N to the structure groupˆ G ⊂ ˆ P . In terms of 2.3, this is a Weyl structure, which is evidently G –invariant.Finally, let ˜ ∇ be the affine connection on ˜ N induced by the Weyl structure above.By construction, ˜ ∇ is G –invariant and torsion–free and, by general principles, itpreserves the underlying geometric structure on ˜ N . (cid:3) In remark 4.3 we have defined the notion of G –symmetry at x ∈ ˜ N . Since any G –symmetry is induced by an element of G , the connection ˜ ∇ constructed aboveis invariant with respect to all G –symmetries. Hence: Corollary.
Let ˜ ∇ be the affine connection on ˜ N = G/H from the Theorem above.Then ( ˜
N , ˜ ∇ ) is an affine symmetric space in the classical sense, whose uniquesymmetry at each point is the G –symmetry. Final remarks. (1) For a reader’s convenience and better orientation in thetext, let us gather all the relations we have discussed from 4.3 till now into thefollowing picture: G × Q ˜ P ˜ P (cid:15) (cid:15) G × H ˆ P ˆ P (cid:15) (cid:15) G +(cid:11) qqqqqqqqqqqq Q & & MMMMMMMMMMMM P (cid:15) (cid:15) G × H ˆ G f f MMMMMMMMMM ˆ G (cid:1) (cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) ˜ M = G/Q (cid:1) (cid:1) (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) & & MMMMMMMMMM ˜ N = G/HM = G/P (2) Note that ˜ N = G/H is a reductive homogeneous space, namely, the reductivedecomposition is g = n ⊕ h where h = g − ⊕ g ⊕ g as above and n = g − ⊕ g .In particular, restriction of the Cartan–Killing form of g to n gives rise to a G –invariant (pseudo–)Riemannian metric on ˜ N whose Levi–Civita connection is thecanonical G –invariant affine connection on G/H = ˜ N . Since both this canonicalconnection and the connection from Theorem 4.4 are invariant with respect to the G –symmetry at each point, so is their difference tensor, which has to vanish bylemma 3.1. Hence the two connections do actually coincide.(3) Note also that for the lowest possible dimension of M , i.e. 3, the dimensionof ˜ N is 4 and the induced almost Grassmannian structure is of type (2 , N is equivalent to the conformal pseudo–Riemannian structure of split signature. Hence the constructions above yield to anexample of non–flat conformal symmetric space in this specific signature.(4) Note finally that the procedure of 4.3 and 4.4 can be applied to any paraboliccontact geometry, however, the resulting structure on the space of chains may differ.For instance, starting with the flat projective contact structure, it turns out thatthe associated path geometry of chains is locally flat, hence it descends to a locallyflat almost Grassmannian structure on the space of chains. On the other hand,the story for CR structures of hypersurface type is completely parallel to that inthe Lagrangean contact case. Of course, understanding the behavior for anotherparabolic contact structures is on the order of the day. References [1] L. Biliotti, On the automorphism group of a second order structure,
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