aa r X i v : . [ m a t h . DG ] J a n SYMMETRIES OF PARABOLIC GEOMETRIES
LENKA ZALABOV ´A
Abstract.
We generalize the concept of affine locally symmetric spaces forparabolic geometries. We discuss mainly | | –graded geometries and we showsome restrictions on their curvature coming from the existence of symmetries.We use the theory of Weyl structures to discuss more interesting | | –gradedgeometries which can carry a symmetry in a point with nonzero curvature.More concretely, we discuss the number of different symmetries which canexist at the point with nonzero curvature. We introduce and discuss symmetries for the so called parabolic geometries. Ourmotivation comes from locally symmetric spaces. Remind that affine locally sym-metric space is a manifold with locally defined symmetries at each point togetherwith an affine connection which is invariant with respect to the symmetries. It canbe proved that the manifold with an affine connection is locally symmetric if andonly if the torsion vanishes and the curvature is covariantly constant. See [11, 9] fordetailed discussion of affine locally symmetric spaces.We are interested in | | –graded parabolic geometries. In this case, the definitionof the symmetry is a generalization of the classical one and follows the intuitiveidea. We show an analogy of the facts known for the affine locally symmetric spacesand we give further results which come from the general theory of parabolic geome-tries. See also [18] for discussion of torsion restrictions for symmetric | | –gradedgeometries.Among all symmetric | | –graded geometries, there are some ‘interesting’ types ofthem, which can carry a symmetry in the point and still allow some nonzero curva-ture at this point. We use the theory of Weyl structures to discuss these geometries,see [6]. In particular, we study the action of symmetries on Weyl structures and viathese actions, we describe the Weyl curvature of the geometries. Finally we discussthe question on the number of different symmetries which can exist in the pointwith nonzero curvature.1. Introduction to Symmetries
The aim of this section is to introduce the definition and discuss basic propertiesof symmetries for parabolic geometries. We first remind some basic definitions andfacts on Cartan and parabolic geometries. In this paper we follow the concepts andnotation of [6, 7] and the reader is advised to consult [4].
Mathematics Subject Classification.
Key words and phrases.
Cartan geometries, parabolic geometries, | | –graded geometries, Weylstructures, symmetric spaces.Discussions with J. Slov´ak, V. ˇZ´adn´ık and A. ˇCap were very useful during the work on thispaper. This research has been supported at different times by the grant GACR 201/05/H005 andby ESI Junior Fellows program. Cartan geometries.
Let G be a Lie group, P ⊂ G a Lie subgroup, and write g and p for their Lie algebras. A Cartan geometry of type (
G, P ) on a smooth manifold M is a principal P –bundle p : G → M together with the 1–form ω ∈ Ω ( G , g ) calleda Cartan connection such that:(1) ( r h ) ∗ ω = Ad h − ◦ ω for each h ∈ P ,(2) ω ( ζ X ( u )) = X for each fundamental vector field ζ X , X ∈ p ,(3) ω ( u ) : T u G −→ g is a linear isomorphism for each u ∈ G .The simplest examples are so called homogeneous models , which are the P –bundles G → G/P endowed with the (left) Maurer Cartan form ω G .The absolute parallelism ω provides the existence of constant vector fields ω − ( X )from X ( G ) defined for all X ∈ g by ω ( ω − ( X )( u )) = X for all u ∈ G . The flow linesof constant vector field ω − ( X ) we denote by Fl ω − ( X ) t ( u ) for u ∈ G .A morphism of Cartan geometries of the same type from ( G →
M, ω ) to ( G ′ → M ′ , ω ′ ) is a principal bundle morphism ϕ : G → G ′ such that ϕ ∗ ω ′ = ω . Furtherwe denote the base morphism of ϕ by ϕ : M → M ′ . Remark that each morphismof Cartan geometries preserves constant vector fields and hence preserves flows ofconstant vector fields, thus T ϕ ◦ ω − ( X ) = ω ′− ( X ) ◦ ϕ,ϕ ◦ Fl ω − ( X ) t ( u ) = Fl ω ′− ( X ) t ( ϕ ( u ))hold for all X ∈ g .We define the kernel of the geometry of type ( G, P ) as the maximal normalsubgroup of G which is contained in P . The geometry is called effective if thekernel is trivial and the geometry is called infinitesimally effective if the kernel isdiscrete. The following Theorem describes useful properties of morphisms, see [14]: Theorem 1.1.
Let ( G →
M, ω ) and ( G ′ → M ′ , ω ′ ) be Cartan geometries of type ( G, P ) and denote by K the kernel. Let ϕ and ϕ be morphisms of these Cartangeometries which cover the same base morphism ϕ : M → M ′ . Then there exists asmooth function f : G → K such that ϕ ( u ) = ϕ ( u ) · f ( u ) for all u ∈ G .In particular, if the geometry is effective, then ϕ = ϕ and f is constant onconnected components of M for infinitesimally effective geometries. We shall mainly deal with automorphisms of Cartan geometries. These are P –bundle automorphisms, which preserve the Cartan connection of the geometry. Inthe homogeneous case, there is the Liouville theorem, see [14]: Theorem 1.2.
Let
G/P be connected. All (locally defined) automorphisms of thehomogeneous model ( G → G/P, ω G ) are left multiplications by elements of G . The structure equation defines the horizontal smooth 2–form K ∈ Ω ( G , g ) inthe following way: K ( ξ, η ) = dω ( ξ, η ) + [ ω ( ξ ) , ω ( η )] . This makes sense for each Cartan connection ω and the form K is called the cur-vature form . Notice that the Maurer–Cartan equation implies that the curvature ofhomogeneous model is zero. It can be proved, see [14]: Theorem 1.3.
If the curvature of a Cartan geometry of type ( G, P ) vanishes, thenthe geometry is locally isomorphic to the homogeneous model ( G → G/P, ω G ) . YMMETRIES OF PARABOLIC GEOMETRIES 3
If the curvature vanishes, then the Cartan geometry is called locally flat . Homo-geneous models are sometimes called flat models.The curvature can be equivalently described by means of the parallelism by the curvature function κ : G → ∧ ( g / p ) ∗ ⊗ g , where κ ( u )( X, Y ) = K ( ω − ( X )( u ) , ω − ( Y )( u )) == [ X, Y ] − ω ([ ω − ( X ) , ω − ( Y )]( u )) . If the values of κ are in ∧ ( g / p ) ∗ ⊗ p , we call the geometry torsion free . We will useboth descriptions of the curvature and we will not distinguish between them. Parabolic geometries.
Let g be a semisimple Lie algebra. A | k | –grading on g isa vector space decomposition g = g − k ⊕ · · · ⊕ g ⊕ · · · ⊕ g k such that [ g i , g j ] ⊂ g i + j for all i and j (we understand g r = 0 for | r | > k ) and suchthat the subalgebra g − := g − k ⊕ · · · ⊕ g − is generated by g − . We suppose thatthere is no simple ideal of g contained in g and that the grading on g is fixed. Eachgrading of g defines the filtration g = g − k ⊃ g − k +1 ⊃ · · · ⊃ g k = g k , where g i = g i ⊕ · · · ⊕ g k . In particular, g and g =: p are subalgebras of g and g =: p + is a nilpotent ideal in p .Let G be a semisimple Lie group with the Lie algebra g . To get a geometry wehave to choose Lie subgroups G ⊂ P ⊂ G with prescribed subalgebras g and p .The obvious choice is this one: G := { g ∈ G | Ad g ( g i ) ⊂ g i , ∀ i = − k, . . . , k } ,P := { g ∈ G | Ad g ( g i ) ⊂ g i , ∀ i = − k, . . . , k } . This is the maximal possible choice, but we may also take the connected componentof the unit in these subgroups or anything between these two extremes. It is notdifficult to show for these subgroups, see [17]:
Proposition 1.4.
Let g be a | k | –graded semisimple Lie algebra and G be a Liegroup with Lie algebra g .(1) G ⊂ P ⊂ G are closed subgroups with Lie algebras g and p , respectively.(2) The map ( g , Z ) g exp Z defines a diffeomorphism G × p + → P . The group P is a semidirect product of the reductive subgroup G and thenilpotent normal subgroup P + := exp p + of P .A parabolic geometry is a Cartan geometry of type ( G, P ), where G and P areas above. If the length of the grading of g is k , then the geometry is called | k | –graded . Parabolic geometries are infinitesimally effective, but they are not effectivein general. Example . Conformal Riemannian structures.
We take the Cartan geometry oftype (
G, P ) where G = O ( p + 1 , q + 1) is the orthogonal group and P is the Poincar´econformal subgroup. Thus the group G is of the form G = n A (cid:12)(cid:12)(cid:12) A (cid:16) J
01 0 0 (cid:17) A − = (cid:16) J
01 0 0 (cid:17)o , LENKA ZALABOV´A where J = (cid:16) E p − E q (cid:17) is the standard product of signature ( p, q ). Its Lie algebra g = o ( p + 1 , q + 1) can be written as g = (cid:26)(cid:18) a Z X A − JZ T − X T J − a (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) a ∈ R , X, Z T ∈ R p + q , A ∈ o ( p, q ) (cid:27) . It leads to a decomposition g = g − ⊕ g ⊕ g , where g − ≃ R p + q , g ≃ R ⊕ o ( p, q )and g ≃ R p + q ∗ . These parts correspond to the block lower triangular part, blockdiagonal part and block upper triangular part and give exactly the | | –grading of g . The elements of the subgroup G = n(cid:16) λ C
00 0 λ − (cid:17) (cid:12)(cid:12)(cid:12) λ ∈ R \ { } , C ∈ O ( p, q ) o preserve this grading. Elements of the subgroup P = G ⋊ exp g preserving thefiltration are of the form (cid:18) λ C
00 0 λ − (cid:19) · (cid:18) Z − ZJZ T E − JZ T (cid:19) = (cid:18) λ λZ − λ ZJZ T C − CJZ T λ − (cid:19) . We get the | | –graded geometry and its homogeneous model is the conformal pseu-dosphere of the corresponding signature.We are mainly interested in the curvature of parabolic geometries and we discusshere the curvature in more detail. It has its values in the cochains C ( g − , g ) for thesecond cohomology H ( g − , g ). This group can be also computed as the homologyof the codifferential ∂ ∗ : ∧ k +1 g ∗− ⊗ g → ∧ k g ∗− ⊗ g . On the decomposable elementsit is given by ∂ ∗ ( Z ∧ · · · ∧ Z k ⊗ W ) = k X i =0 ( − i +1 Z ∧ · · · ˆ i · · · ∧ Z k ⊗ [ Z i , W ]+ X i 0. The crucial structural description of the curvature is providedby the following Theorem, see [17]: YMMETRIES OF PARABOLIC GEOMETRIES 5 Theorem 1.6. The curvature κ of a regular normal parabolic geometry vanishes ifand only if its harmonic part κ H vanishes. Moreover, if all homogeneous componentsof κ of degrees less than j vanish identically and there is no cohomology H j ( g − , g ) ,then also the curvature component of degree j vanishes. Another possibility is the decomposition of the curvature according to the values: κ = k X j = − k κ j . In an arbitrary frame u we have κ j ( u ) ∈ g − ∧ g − → g j . The component κ − valuedin g − is exactly the torsion of the geometry.Remark that in the | | –graded case the decomposition by the homogeneity corre-sponds to the decomposition according to the values. The homogeneous componentof degree 1 coincides with the torsion while the homogeneous components of degrees2 and 3 correspond to κ and κ . Thus all | | –graded geometries are clearly regular. Underlying structures for parabolic geometries and tractor bundles. Itis well known, that the existence of the Cartan connection allows to describe thetangent bundle of the base manifold as the associated bundle T M ≃ G × P g / p ,where the action of P on g / p is induced by the Ad–action on g . In the case ofparabolic geometry, we can use the usual identification g / p ≃ g − . Thus we canwrite T M ≃ G × P g − for the so called Ad–action of P on g − , which is obviouslygiven e.g. by the condition Ad = π ◦ Ad, where π is the projection g → g − in thedirection of p . In particular, if we denote p : G → M the projection, then one cansee from the properties of Cartan connection that the mapping G × g − → T M givenby ( u, X ) T p.ω − ( X )( u ) factorizes to the bundle isomorphism T M ≃ G × P g − .We use the notation [[ u, X ]] for the tangent vectors and more generally, we denoteelements and sections of arbitrary asociated bundle by the same symbol.Remark that vector fields can be equivalently understand as P –equivariant map-pings G → g − , so called frame forms . Each frame form s describes the vector[[ u, s ( u )]] ∈ T p ( u ) M . We have similar identifications for the cotangent bundle andarbitrary tensor bundles.The morphism ϕ of Cartan geometries induces uniquely by means of its basemorphism ϕ the tangent morphism T ϕ : T M → T M ′ and it can be written usingthe previous identification as T ϕ ([[ u, X ]]) = [[ ϕ ( u ) , X ]] . Again, similar fact works for the cotangent bundle and arbitrary tensor bundlesand can be rewritten in the language of frame forms. In the sequel, we will use bothdescriptions of tensors and tensor fields.There is also a different concept of interesting associated bundles available forall Cartan geometries. First, we can define the adjoint tractor bundle A M as theassociated bundle A M := G × P g with respect to the adjoint action. The Lie bracketon g defines a bundle map { , } : A M ⊗ A M → A M which makes any fiber of A M into a Lie algebra isomorphic to the Lie algebra g .For all u ∈ G and X, Y ∈ g the bracket is defined by { [[ u, X ]] , [[ u, Y ]] } = [[ u, [ X, Y ]]] . LENKA ZALABOV´A More generally, let λ : G → Gl ( V ) be a linear representation. We define the tractor bundle V M as the associated bundle V M := G × P V with respect to therestriction of the action λ to the subgroup P . In the case of the adjoint representa-tion Ad : G → Gl ( g ) we get exactly the adjoint bundle. Elements of the associatedbundle and also the sections of the bundle are called ( adjoint ) tractors .These bundles allow us to describe nicely structures underlying parabolic ge-ometries and descriptions via them will be useful for some computations later. Wewill not devote the whole theory. We will only shortly remind basic facts whichallow us to use tractor language in the future. Note that one has to start with therepresentation of the whole G to define tractor bundles. There are differences be-tween properties of tractor bundles and e.g. tangent bundles. See [3, 4] for detaileddiscussion.The adjoint tractor bundle A M acts on an arbitrary tractor bundle V M = G × P V . For all u ∈ G , X ∈ g and v ∈ V we define the algebraic action • : A M ⊗ V M → V M, [[ u, X ]] • [[ u, v ]] = [[ u, λ ′ X ( v )]] , where λ ′ : g → gl ( V ) is the infinitesimal representation given by λ : G → Gl ( V ).In fact, any G –representation λ gives rise to the natural bundle V on Cartangeometries of type ( G, P ) and all P –invariant operations on representations giverise to geometric operations on the corresponding natural bundles. Similarly, the P –equivariant morphisms of representations give corresponding bundle morphisms.In this way, the projection π : g → g / p naturally induces the bundle projectionΠ : G × P g = A M −→ T M = G × P g / p and we can see the adjoint tractor bundle as an extension of the tangent bundle.Now, suppose that we have a | k | –graded parabolic geometry. The filtration of g is P –invariant and induces the filtration of the adjoint subbundles A M = A − k M ⊃ A − k +1 M ⊃ · · · ⊃ A k M, where A i M := G × P g i . At the same time, we get the associated graded bundle gr( A M ) = G × P gr( g ) = A − k M ⊕ A − k +1 M ⊕ · · · ⊕ A k M, where A i M = A i M/ A i +1 M ≃ G × P g i / g i +1 . Since the Lie bracket on gr( g ) is P + –invariant, there is the algebraic bracket on gr( A M ) defined by means of the Liebracket. This bracket is compatible with the latter bracket on the adjoint tractorbundle and we denote both brackets by the same symbol. We have { , } : A i M × A j M → A i + j M. From above we see that T M ≃ A M/ A M and we obtain the induced filtrationof the tangent bundle T M = T − k M ⊃ T − k +1 M ⊃ · · · ⊃ T − M, where T i M ≃ A i M/ A M . Again, the filtration of T M gives rise to the associatedgraded bundle gr( T M ) = gr − k ( T M ) ⊕ · · · ⊕ gr − ( T M ) , where gr i ( T M ) = T i M/T i +1 M ≃ A i M. The action of P + on G is free and thequotient G /P + =: G → M is a principal bundle with structure group G = P/P + .The action of P + on g i / g i +1 is trivial and we have g i / g i +1 ≃ g i as G –modules.We get gr i ( T M ) ≃ G × G g i and gr( T M ) ≃ G × G g − . For each x ∈ M , the spacegr( T x M ) is the nilpotent graded Lie algebra isomorphic to the algebra g − . YMMETRIES OF PARABOLIC GEOMETRIES 7 Now, we would like to describe the structure underlying parabolic geometries.These are nicely related to filtered manifolds. Remark, that a filtered manifold is amanifold M together with a filtration T M = T − k M ⊃ · · · ⊃ T − M such that forsections ξ of T i M and η of T j M the Lie bracket [ ξ, η ] is a section of T i + j M . Onthe corresponding associated graded bundle we obtain the Levi bracket L : gr( T M ) × gr( T M ) → gr( T M ) , which is induced by T i M × T j M → gr i + j ( T M ), the composition of the Lie bracketof the latter vector fields with the natural projection T i + j M → gr i + j ( T M ). Thisdepends only on the classes in gr i ( T M ) and gr j ( T M ) and gives a mapgr i ( T M ) × gr j ( T M ) → gr i + j ( T M ) . For each x ∈ M , this makes gr( T x M ) into a nilpotent graded Lie algebra.For each parabolic geometry, we have described some canonical filtration of T M induced from the grading of g . It can be proved that geometry is regular if andonly if this induced filtration of T M makes M into the filtered manifold such thatthe latter bracket on each gr( T x M ) coincides with { , } . In particular, we havegr( T x M ) ≃ g − for each x ∈ M , see [7].If we start with a regular parabolic geometry, we get exactly the following dataon the base manifold: • A filtration { T i M } of the tangent bundle such that gr( T x M ) ≃ g − for each x ∈ M . • A reduction of structure group of the associated graded bundle gr( T M )with respect to Ad : G → Aut gr ( g − ). (The reduction is trivial in the case G = Aut gr ( g − ).)These data are called the underlying infinitesimal flag structure . The proof of thefollowing equivalence between such infinitesimal flag structures and regular normalparabolic geometries can be found in [17, 5]: Theorem 1.7. Let M be a filtered manifold such that gr( T x M ) ≃ g − for each x ∈ M and let G → M be a reduction of gr( T M ) to the structure group G . Thenthere is a regular normal parabolic geometry ( p : G → M, ω ) inducing the givendata. If H ℓ ( g − , g ) are trivial for all ℓ > then the normal regular geometry isunique up to isomorphism. The construction is functorial and the latter Theorem describes an equivalenceof categories.In the case of | | –graded geometries, the filtration of the tangent bundle is trivial.We need only the reduction of gr( T M ) to the structure group G . The | | –gradedgeometries are automatically regular and we get the correspondence between nor-mal | | –graded parabolic geometries and first order G–structures with structuregroup G . Symmetries and homogeneous models. In general, automorphisms of Cartangeometries have to preserve the underlying structure given by the existence of theCartan connection. Thus in the case of parabolic geometries, automorphisms espe-cially preserve the induced filtration of the manifold. In the | | –graded case, theycorrespond to the automorphisms of G –structures. We define the symmetry on theparabolic geometry in the following way: LENKA ZALABOV´A Definition 1.8. Let ( G → M, ω ) be a parabolic geometry. The ( local ) symmetrywith the center at x ∈ M is a (locally defined) diffeomorphism s x on M 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 ϕ of the Cartan geometry, i.e. s x = ϕ (on some neighborhood of x ).The geometry is called ( locally ) symmetric if there is a symmetry at each x ∈ M .In this paper, we will discuss only locally defined symmetries and locally symmet-ric geometries. We will omit the word ‘locally’ and we will shortly say ‘symmetry’and ‘symmetric geometry’. The relation between locally and globally defined sym-metries we will discuss elsewhere.In other words, symmetries revert by the sign change only the smallest subspacein the filtration, while their actions on the rest are completely determined by thealgebraic structure of g and p . To see that it is not reasonable to define the symmetryas an automorphism such that its differential reverts the whole tangent space, itsuffice to discuss the filtration and brackets on | | –graded geometries, see [18]. Seealso [10] for discussion of Cauchy–Riemann structures.However, we are interested in | | –graded parabolic geometries. In this case, thefiltration is trivial and thus T − M = T M . First two properties in the definitionthan say that symmetries follow completely the classical intuitive idea. The thirdone gives that we can understand the latter defined symmetries as symmetries ofthe corresponding G –structure (and thus of the corresponding geometry).The class of | | –graded geometries involves many well known types of geometrieslike conformal and projective structures. There are known generalizations of affinelocally symmetric spaces to concrete examples of them, see e.g. [13] for discussion ofthe projective case in the classical setup. We use the universal language for parabolicgeometries to study the properties of all symmetric | | –graded geometries together.We start the discussion with one useful observation, see also [18]: Lemma 1.9. If there is a symmetry with the center at x on a | | –graded geometryof type ( G, P ) , then there exists an element g ∈ P such that Ad g ( X ) = − X for all X ∈ g − , where Ad denotes the action on g − induced from the adjoint action.All these elements are of the form g = g exp Z , where g ∈ G such that Ad g ( X ) = − X for all X ∈ g − and Z ∈ g is arbitrary.Proof. Let ϕ cover some symmetry with the center at x ∈ M . The symmetry ϕ preserves the point x and hence the morphism ϕ preserves the fiber over x . Let u be an arbitrary fixed point in the fiber over x . There is an element g ∈ P such that ϕ ( u ) = u · g . We show that this g satisfies the condition.Let ξ ∈ X ( M ) be a vector field on M . In the point x we have T ϕ.ξ ( x ) = − id T x M ( ξ ( x )) = − ξ ( x ) . We can also write ξ ( x ) = [[ u, X ]] for a suitable X ∈ g − . The symmetry simplychanges the sign of the coordinates X in the chosen frame u and we get T ϕ ([[ u, X ]]) = [[ u, − X ]] . In the fiber over x , the equivariancy gives T ϕ ([[ u, X ]]) = [[ ϕ ( u ) , X ]] = [[ ug, X ]] = [[ u, Ad g − ( X )]] . YMMETRIES OF PARABOLIC GEOMETRIES 9 Comparing the coordinates in the frame u gives us the action of element g ∈ P on g − . We have Ad g ( − X ) = X and thus the action of element g is the change of thesign for all elements from g − .Next, because g ∈ P , we have g = g exp Z for some g ∈ G and Z ∈ g . Wehave Ad g exp Z ( X ) = − X for all X ∈ g − . But the action of the component exp Z is trivial while the action of g preserves the grading, i.e. Ad g = Ad g , and theelement g satisfies Ad g ( X ) = − X . (cid:3) As a consequence of the proof of the Lemma 1.9 and Theorem 1.2 we get thefollowing conditions for homogeneous models: Proposition 1.10. (1) All symmetries of the homogeneous model ( G → G/P, ω G ) at the origin o are exactly the left multiplications by elements g ∈ P satisfying thecondition in the Lemma 1.9.(2) If there is a symmetry at the origin o , then the homogeneous model is symmetricand there is an infinite amount of symmetries at each point.(3) If there is no such element, then none Cartan geometry of the same type carrysome symmetry. The Proposition gives us nice and simple criterion to decide, whether the | | –graded geometry of given type allow some symmetric geometry or not. Example . Projective structures. We can make two reasonable choices of theLie group G with the | | –graded Lie algebra sl ( m + 1 , R ). We can consider G = Sl ( m + 1 , R ). Then the maximal P is the subgroup of all matrices of the form (cid:0) d W D (cid:1) such that d = det D for D ∈ Gl ( m, R ) and W ∈ R m ∗ , but we take only theconnected component of the unit. It consists of all elements such that det D > G acts on rays in R m +1 and P is the stabilizer of theray spanned by the first basis vector. Clearly, with this choice G/P is diffeomorphicto the m –dimensional sphere. The subgroup G contains exactly elements of P suchthat W = 0, and this subgroup is isomorphic to Gl + ( m, R ).The second reasonable choice is G = P Gl ( m + 1 , R ), the quotient of Gl ( m + 1 , R )by the subgroup of all multiples of the identity. This group acts on R P m and asthe subgroup P we take the stabilizer of the line generated by the first basis vector.Clearly G/P is diffeomorphic to R P m . The subgroup G is isomorphic to Gl ( m, R ),because each class in G has exactly one representant of the form ( D ).We can make the computation simultaneously and then discuss the cases sepa-rately. We have g = n(cid:0) − tr ( A ) ZX A (cid:1) (cid:12)(cid:12)(cid:12) X, Z T ∈ R m , A ∈ gl ( m, R ) o and elements from g − look like ( X ). For each a = (cid:0) b B (cid:1) ∈ G and V = ( X ) ∈ g − , the ad-joint action Ad a V is given by X b − BX . We look for elements (cid:0) b B (cid:1) such that BX = − bX for each X . It is easy to see that B is a diagonal matrix and that allelements on the diagonal are equal to − b . Thus we may represent the prospectivesolution as (cid:0) − E (cid:1) .Now we discuss the choice G = Sl ( m + 1 , R ) with G/P ≃ S m . The elementhas the determinant ± m is even, then the element gives a symmetry but if m is odd, then there is nosymmetry on this model. In the case of G = P Gl ( m + 1 , R ), the above elementalways represents the class in G and thus yields the symmetry. In both cases, allelements giving a symmetry in the origin are of the form (cid:0) W − E (cid:1) for all W ∈ R m ∗ . These two choices of the groups G and P correspond to projective structures onoriented and non–oriented manifolds. The non–oriented projective geometries canbe symmetric, the homogeneous models are symmetric. The existence of a sym-metry on the oriented projective geometry depends on its dimension. Clearly, thehomogeneous model is the oriented sphere with the canonical projective structure(represented e.g. by the metric connection of the round sphere metric) and the ob-vious symmetries are orientation preserving in the even dimensions only. Thus onlythe even–dimensional geometries can be symmetric. Symmetric odd–dimensionaloriented projective geometry does not exist.There is the well known classification of semisimple Lie algebras in terms ofsimple roots and for a given g , there is a complete description of all parabolicsubalgebras, see [17, 7] for more details. This allows to describe all correspondingparabolic geometries (with simple Lie group), see [6, 18] for the list. One can seefrom the latter example that for concrete | | –graded geometry, it depends mainlyon the choice of groups G and P with the corresponding Lie algebras whether thehomogeneous model of corresponding | | –graded geometry is symmetric.For the geometries from the list, there are standard choices of groups whichgive well know geometries. One can simply take and discuss this choices for eachgeometry from the list. We saw this discussion in the projective case (the case G = Sl ( m, R )) and one can also see Example 3.8 for the conformal case. In theother cases, the discussion is analogous. The other possibility is to ask, whetherthere is a choice of groups with the given g and p such that the correspondingmodel is symmetric. Again, we saw this approach in the latter Example (the case G = P Gl ( m, R )).We remind here only the best known examples, where is the existence of sym-metries more or less clear. They are also the most interesting ones. See [18] fordiscussion of all | | –graded cases. Theorem 1.12. Homogeneous models of the following | | –graded parabolic geome-tries are symmetric: • almost Grassmannian geometries modeled over the Grassmannians of p -planes in R p + q (i.e. of type ( p, q ) ) where p, q ≥ , G = P Gl ( p + q, R ) , • projective geometries in dim ≥ , G = P Gl ( m + 1 , R ) , • conformal geometries in all signatures in dim ≥ , G = O ( p + 1 , q + 1) , • almost quaternionic geometries, G = P Gl ( m + 1 , H ) . In other words, the latter geometries admit symmetric space – the flat space G/P .2. Torsion restrictions Motivated by the affine case, we find some restriction on the curvature of | | –graded geometry carrying some symmetry. See [11, 9] for discussion of classicalaffine locally symmetric spaces in more detail. We study the curvature of parabolicgeometries in a similar way. We also give some corollaries of the results comingfrom the general theory of parabolic geometries. Locally flat geometries. Remark that the curvature of a | | –graded geometry isdescribed by the curvature function κ : G → ∧ g ∗− ⊗ g and the torsion is identifiedwith the part κ − . This is correctly defined component of the curvature, we havejust to keep in mind the proper action of P on g − . YMMETRIES OF PARABOLIC GEOMETRIES 11 The following Proposition is the analogy of the classical result for the affinelocally symmetric spaces, see also [18]. Proposition 2.1. Symmetric | | –graded parabolic geometries are torsion free.Proof. Choose an arbitrary x ∈ M on a symmetric | | –graded geometry of type( G, P ) and let ϕ cover a symmetry with the center at x . The symmetry ϕ preserves x and thus ϕ preserves the fiber over x . The curvature function satisfies κ = κ ◦ ϕ and we have κ ( u ) = κ ( ϕ ( u )) = κ ( u · g ) = g − · κ ( u )for suitable g ∈ P and the same holds for κ − . We compare κ − in the frames u and ϕ ( u ). Because g is exactly the element from Lemma 1.9 which acts on g − as − id, we have κ − ( ϕ ( u ))( X, Y ) = κ − ( u · g )( X, Y ) = g − · κ − ( u )( X, Y ) == Ad g − ( κ − ( u )(Ad g X, Ad g Y )) == − κ − ( u )( − X, − Y ) = − κ − ( u )( X, Y ) . This is equal to κ − ( u )( X, Y ) and we obtain κ − ( u )( X, Y ) = − κ − ( u )( X, Y ) forall X, Y ∈ g − . Thus κ − ( u ) vanishes and this holds for all frames u ∈ G over x .The torsion then vanishes at x .If the geometry is symmetric, then there is some symmetry at each x ∈ M . Then κ − vanishes for all x ∈ M and the geometry is torsion free. (cid:3) The theory on harmonic curvature allows us to prove stronger restriction formany types of | | –graded parabolic geometries. See [17, 6] for detailed discussion ofthe theory. As a corollary of Theorem 1.6 and Proposition 2.1 we get the followingProposition, see [18]: Proposition 2.2. Let ( G → M, ω ) be a normal | | –graded parabolic geometry suchthat its homogeneous components of the harmonic curvature are only of degree .If there is a symmetry at a point x ∈ M , then the whole curvature vanishes in thispoint. In particular, if the geometry is symmetric than it is locally isomorphic withthe homogeneous model. The harmonic curvature can be defined only for regular normal geometries. Itis much simpler object then the whole curvature and the main feature is that it ispossible to find algorithmically all its components.From the point of view of underlying structures, the condition ‘normal’ is as-sumption of technical character. For an appropriate G –structure on the manifoldwe can find | | –graded normal parabolic geometry inducing given structure. If wetake symmetries as morphisms of the concrete underlying geometry, this assumptiongives us no restriction. Remark . It is easy to see, that similar arguments apply in the case when theonly homogeneous components are of degree 3, see Theorem 1.6. Clearly, the part κ vanishes for the same reason as the torsion.For all parabolic geometries, one can compute the corresponding components ofthe harmonic curvature. Computation of the cohomology of (complex) Lie algebrasis based on Kostant’s version of the Bott–Borel–Weil theorem and the algorithmcan be found in [15, 16]. This allows us to list explicitely, which geometries satisfythe above condition on the harmonic curvature and which not. One gets that most of | | –graded geometries has homogeneous components only of degree 1 and hasto be locally flat, if they are symmetric. See [18] for discussion of all | | –gradedgeometries. Only geometries from the list in Theorem 1.12 are more interesting.Some of them satisfy conditions from the Proposition 2.2 or Remark 2.3: Corollary 2.4. Symmetric normal | | –graded geometries of the following types arelocally flat: • almost Grassmannian geometries such that p > and q > , • conformal geometries in all signatures of dimension , • projective geometries of dimension . The other geometries from the Theorem 1.12 are the only | | –graded geometrieswhich can carry symmetries in the points with nonzero curvature. Theorem 2.5. The following normal | | –graded geometries can admit a symmetricspace, which is not locally isomorphic to the homogeneous model of the same type: • projective geometries of dim > , • conformal geometries in all signatures of dim > , • almost quaternionic geometries, • almost Grassmannian structures such that p = 2 or q = 2 . The main property of this geometries is that they allow some homogeneouscomponent of harmonic curvature of degree 2. We have no restriction for thesegeometries coming from the theory of harmonic curvature.3. Weyl Structures and Symmetries We study the action of symmetries on so called Weyl structures, which providemore convenient understanding of the underlying structure on the manifold. See[6] for more detailed discussion on Weyl structures. We also discuss symmetries oneffective geometries in a little more detail. Weyl structures. Weyl structures are our main tool to deal with the interestinggeometries. They exist for all parabolic geometries. We describe Weyl structuresonly in the | | –graded case, general theory can be found in [6, 7].Remind that there is the underlying bundle G := G / exp g for each | | –gradedgeometry, which is the principal bundle p : G → M with structure group G .At the same time we get the principal bundle π : G → G with structure group P + = exp g .The Weyl structure σ for a | | –graded geometry ( p : G → M, ω ) is a globalsmooth G –equivariant section of the projection π : G → G . There exists someWeyl structure σ : G → G for each | | –graded geometry and for arbitrary twoWeyl structures σ and ˆ σ , there is exactly one G –equivariant mapping Υ : G → g such that ˆ σ ( u ) = σ ( u ) · exp Υ( u )for all u ∈ G . The equivariancy allows to extend Υ to P –equivariant mapping G → g and in fact, it is a 1–form on M . Weyl structures form an affine space modeledover the vector space of all 1–forms and in this sense we can write ˆ σ = σ + Υ.The choice of the Weyl structure σ induces the decomposition of all tractorbundles into G –invariant pieces. In particular, the adjoint tractor bundle splits as A M = T M ⊕ End ( T M ) ⊕ T ∗ M, YMMETRIES OF PARABOLIC GEOMETRIES 13 which is compatible with the latter facts on underlying structures. Thus End ( T M )is the appropriate subbundle of End( T M ) and the algebraic bracket of a vector fieldwith a 1–form becomes an endomorphism on T M .The choice of the Weyl structure σ also defines the decomposition of the 1–form σ ∗ ω ∈ Ω ( G , g ) such that σ ∗ ω = σ ∗ ω − + σ ∗ ω + σ ∗ ω . The part σ ∗ ω ∈ Ω ( G , g ) defines the principal connection on p : G → M ,the Weyl connection . For each linear representation V we get the induced Weylconnection ∇ σ on G × λ V and for arbitrary two Weyl structures σ and ˆ σ = σ · exp Υ,there is the explicit formula for the change of corresponding connections ∇ σ and ∇ ˆ σ . For a vector field ξ ∈ X ( M ) and a section s of G × λ V we have ∇ ˆ σξ ( s ) = ∇ σξ ( s ) + { ξ, Υ } • s. The algebraic bracket of a vector field with a 1–form becomes an endomorphism on T M and • is just the algebraic action derived from λ .Among general Weyl structures, there is an interesting class of them which arecrucial in the sequel – the normal Weyl structures. We define the normal Weylstructure at u as the only G –equivariant section σ u : G → G satisfying σ u ◦ π ◦ Fl ω − ( X )1 ( u ) = Fl ω − ( X )1 ( u ) . Clearly, normal Weyl structures are defined locally over some neighborhood of p ( u )and can be used only for discussion of local properties. They are closely related tothe normal coordinate systems for parabolic geometries and generalize the affinenormal coordinate systems, see [7, 8, 21].Finally, let us shortly concentrate on automorphisms. For each automorphism ϕ of the geometry, there is the pullback ϕ ∗ σ = ϕ − ◦ σ ◦ ϕ of the Weyl structure σ ,where ϕ is the underlying automorphism on G induced by ϕ . This is again someWeyl structure and there is exactly one Υ such that ϕ ∗ σ = σ + Υ . In addition, thepullback respects the affine structure, i.e. ϕ ∗ ( σ + Υ) = ϕ ∗ σ + ϕ ∗ Υ . There is also a crucial fact, that the pullback of normal Weyl structure is againnormal Weyl structure. It is easy to see it from the following computation: ϕ ∗ σ u ◦ π ◦ Fl ω − ( X )1 ( u ) = ϕ − ◦ σ u ◦ ϕ ◦ π ◦ Fl ω − ( X )1 ( u ) == ϕ − ◦ σ u ◦ π ◦ Fl ω − ( X )1 ( ϕ ( u )) == ϕ − ◦ ϕ ◦ Fl ω − ( X )1 ( u ) = Fl ω − ( X )1 ( u ) . Thus the pullback ϕ ∗ σ u again satisfies the conditions on normal Weyl structures. Actions of symmetries on Weyl structures. Let us return to the | | –gradedgeometries carrying some symmetries. We discuss the action of coverings of sym-metries on Weyl structures. Proposition 3.1. Let ( G → M, ω ) be a | | –graded geometry and suppose there isa symmetry with the center at x ∈ M covered by some automorphism ϕ . There isa Weyl structure σ such that ϕ ∗ σ | p − ( x ) = σ | p − ( x ) . Thus in the fiber over x , the pullback of this σ along ϕ equals to the same Weylstructure σ .Proof. Choose an arbitrary Weyl structure ˆ σ and compute the pullback of thisstructure along ϕ . The result is another Weyl structure ϕ ∗ ˆ σ = ˆ σ + Υ . We provethat the Weyl structure ˆ σ + Υ satisfies the condition. We compute ϕ ∗ (ˆ σ + 12 Υ) = ϕ ∗ ˆ σ + ϕ ∗ 12 Υ = ˆ σ + Υ + 12 ϕ ∗ Υand it suffices to show that ϕ ∗ Υ( u ) = (Υ ◦ ϕ )( u ) = − Υ( u ) holds for u ∈ p − ( x ) ⊂G . Clearly, ϕ : G → G preserves p − ( x ) and in fact, it is equal to the right actionof some suitable element from G . This is exactly the element g from Lemma 1.9corresponding to the frame u . Thanks to the equivariancy and the fact that thevalues of Υ are in g (the dual of g − ), the action of the element changes the signand we get (Υ ◦ ϕ )( u ) = Υ( ug ) = − Υ( u ).At the point x the latter fact givesˆ σ + Υ + 12 ϕ ∗ Υ = ˆ σ + Υ − 12 Υ = ˆ σ + 12 Υ . If we put all together we get ϕ ∗ (ˆ σ + Υ) = ˆ σ + Υ at x and thus the action of ϕ preserves the Weyl structure σ := ˆ σ + Υ in the fiber over x ∈ M . (cid:3) In the sequel, we call any such Weyl structure invariant with respect to ϕ at x or shortly ϕ –invariant at x .There can be more than one invariant Weyl structure at x with respect to thesame ϕ , but all of them have to coincide at x . Let σ, ¯ σ be different ϕ –invariant Weylstructures at x . We know that ¯ σ = σ + Υ for some in general nonzero Υ : G → g .At the point x we get¯ σ = ϕ ∗ ¯ σ = ϕ ∗ ( σ + Υ) = ϕ ∗ σ + ϕ ∗ Υ = σ + ϕ ∗ Υ . The relation σ + Υ = σ + ϕ ∗ Υ implies Υ = ϕ ∗ Υ at x . Because ϕ ∗ Υ = − Υ holds inthe fiber over x we get the vanishing of Υ at x . In general, we know nothing aboutthe neighborhood of x .Remark that if σ is ϕ –invariant Weyl structure at x then ϕ ∗ σ is again ϕ –invariantat x because ϕ ∗ σ = σ implies ϕ ∗ ( ϕ ∗ σ ) = ϕ ∗ σ at x . We can shortly say, that thepullback along ϕ permutes all Weyl structures invariant with respect to ϕ at x . Inaddition, we can prove the following Theorem. Theorem 3.2. Suppose that ϕ covers some symmetry with the center at x on a | | –graded geometry. There is exactly one normal Weyl structure σ u such that ϕ ∗ σ u = σ u over some neighborhood of the center x .Proof. Let σ be an arbitrary ϕ –invariant Weyl structure at x , i.e. ϕ ∗ σ = σ holdsin the fiber over x . We take the normal Weyl structure σ u such that σ u ( v ) = σ ( v )for p ( v ) = x . The condition of normality prescribes it then uniquely on a suitableneighborhood of x ∈ M . Pullback of this Weyl structure is again some normal Weylstructure. But we know that σ and σ u coincide at x and we have ϕ ∗ σ u = σ u at x .Then ϕ ∗ σ u has to be the original normal Weyl structure σ u and we get ϕ ∗ σ u = σ u over some neighborhood of x . YMMETRIES OF PARABOLIC GEOMETRIES 15 Finally, the resulting normal Weyl structure does not depend on the choice of the ϕ –invariant Weyl structure σ at x , because all these structures are equal at x . (cid:3) Corollary 3.3. Suppose there is a symmetry with the center at x on a | | –gradedgeometry. Then there is an admissible affine connection given on the neighborhoodof x which is invariant with respect to the symmetry.Proof. We take the Weyl connection ∇ σ u given by the normal Weyl structure invari-ant on the neighborhood of x with respect to some covering of the symmetry. (cid:3) Effective geometries. Let us pass our discussion to the relation of various cover-ings of the symmetry at x and their invariant Weyl structures at x . We also discusshere effective geometries in more detail. We start with a useful Lemma. Lemma 3.4. (1) The kernel K of the parabolic geometry of type ( G, P ) is exactlythe kernel of the adjoint action Ad : G → Gl ( g ) . In particular, it is contained in G .(2) Let φ be an automorphism of a | | –graded geometry such that its base morphism φ preserves some x ∈ M . If φ ( u ) = u · h for some h ∈ K and for some u ∈ p − ( x ) (thus for all u ∈ p − ( x ) ), then φ = id M on some neighborhood of x .Proof. (1) Remind that K is the maximal normal subgroup of G contained in P . Suppose that g belongs to the kernel K . We defined parabolic geometries asinfinitesimally effective geometries and K is discrete in this case. Then thanks tothe smoothness of the multiplication we have exp( tX ) g exp( − tX ) = g for all X ∈ g and the differentiating at t = 0 gives T ρ g .X − T λ g .X = 0. Thus Ad g ( X ) = X holdsfor all X ∈ g and the element g lies in the kernel of the adjoint action.Suppose that g belongs to the kernel of the adjoint action Ad : G → Gl ( g ). ThenAd g : g → g is identity or equivalently Ad g ( X ) = X for all X ∈ g . If h ∈ G isarbitrary then Ad hgh − ( X ) = Ad h Ad g Ad h − ( X ) = Ad h (Ad h − ( X )) = X for all X ∈ g . The action of g clearly respects the grading and g belongs to G ⊂ P . Thuselements from the kernel of the adjoint action form a normal subgroup of G whichis contained in P . Such subgroup has to be contained in K .(2) We have φ ( x ) = x and we use normal coordinates at u ∈ p − ( x ) to describe theneighborhood of x . Any point from the suitable neighborhood of x can be writtenas p ◦ Fl ω − ( X )1 ( u ) for suitable X ∈ g − and we have φ ◦ p ◦ Fl ω − ( X )1 ( u ) = p ◦ φ ◦ Fl ω − ( X )1 ( u ) = p ◦ Fl ω − ( X )1 ( φ ( u )) == p ◦ Fl ω − ( X )1 ( uh ) . The equivariancy of ω and the fact that Ad h = Ad h for h ∈ G give that thecurve p ◦ Fl ω − ( X ) t ( uh ) coincides with the curve p ◦ Fl ω − (Ad h − X ) t ( u ). (See [8, 21]for details on generalized geodesics.) The action of h from the kernel is trivial andthen the action of h − is trivial, too. We have p ◦ Fl ω − ( X )1 ( uh ) = p ◦ Fl ω − (Ad h − X )1 ( u ) = p ◦ Fl ω − ( X )1 ( u ) . This holds for all X ∈ g − and φ is the identity on a neighborhood of x . (cid:3) Proposition 3.5. Let ϕ and ψ be two coverings of two symmetries with the centerat x on a | | –graded geometry.(1) Suppose that invariant Weyl structures with respect to this two coverings coin-cide at x , i.e. ϕ ∗ σ | p − ( x ) = σ | p − ( x ) = ψ ∗ σ | p − ( x ) . Then ψ ( u ) = ϕ ( u ) · h holds for all u over some neighborhood of x and for some h from the kernel of the geometry.In particular, ϕ and ψ have to cover the same symmetry with center at x .(2) Suppose that ϕ and ψ cover the same symmetry at x . Then Weyl structuresinvariant with respect to this two coverings coincide along the fiber over x .Proof. (1) Suppose that ϕ ∗ σ = σ and ψ ∗ σ = σ at x . Then we have ϕ − ◦ σ ◦ ϕ = ψ − ◦ σ ◦ ψ at x and this is equivalent with the fact that( ψ ◦ ϕ − ) ◦ σ ( v ) = σ ◦ ( ψ ◦ ϕ − )( v )holds for each v ∈ p − ( x ). The morphism ϕ preserves the fiber over x and in fixed v is equal to the right multiplication by some k ∈ G such that Ad k ( X ) = − X forall X ∈ g − . The morphism ψ also coincides at v with the action of some g ∈ G satisfying Ad g ( X ) = − X for all X ∈ g − . We can accordingly write( ψ ◦ ϕ − ) ◦ σ ( v ) = σ ( v ) · h, where h = k − g acts trivially on g − by the Ad–action. Such element h ∈ G has toact trivially on the whole g because the action respects the grading and the actionon g ⊆ g ∗− ⊗ g − and g ≃ g ∗− is trivial. Thus h belongs to the kernel and then ψ ◦ ϕ − equals to the identity on some neighborhood of x ∈ M , see Lemma 3.4.Then Theorem 1.1 gives that ψ ◦ ϕ − ( u ) = id G · f ( u )holds for some function f : G → K over the neighborhood of x . According to ourdefinition of parabolic geometries, they are always infinitesimally effective and thefunction f has to be constant on the neighborhood of x . The value of f has to be thelatter element h ∈ G . We have ψ ◦ ϕ − ( u ) = uh and if we apply the automorphism ϕ first, we get ψ ◦ ϕ − ( ϕ ( u )) = ϕ ( u ) h over a suitable neighborhood of x . Thisimplies ψ ( u ) = ϕ ( u ) · h over the neighborhood of x .(2) We have ψ ( u ) = ϕ ( u ) · h over some suitable neighborhood of x , where h belongsto the kernel, see Theorem 1.1. Then clearly their underlying automorphisms on G satisfy ψ ( v ) = φ ( v ) · h for v ∈ G . Suppose that ϕ ∗ σ ( v ) = ϕ − ◦ σ ◦ ϕ ( v ) = σ ( v )for all v ∈ p − ( x ) and the Weyl structure σ . We have ψ ∗ σ ( v ) = ( ψ − ◦ σ ◦ ψ )( v ) = ϕ − ( σ ◦ ψ ( v )) · h − == ( ϕ − ◦ σ ◦ ϕ )( v ) · h · h − = ϕ ∗ σ ( v ) = σ ( v )for all v ∈ p − ( x ) and the invariant Weyl structures coincide at x . (cid:3) In particular, if ϕ covers some symmetry at x , then clearly ϕ − covers some sym-metry at x , too. Moreover, ϕ –invariant and ϕ − –invariant Weyl structures coincideat x . Really, if we have ϕ ∗ σ = σ at x then we also have ( ϕ − ) ∗ ϕ ∗ σ = ( ϕ − ) ∗ σ at x and simultaneously we get( ϕ − ) ∗ ϕ ∗ σ = ( ϕ ◦ ϕ − ) ∗ σ = id ∗ σ = σ. Thus we get ( ϕ − ) ∗ σ = σ at x . Then ϕ and ϕ − have to cover the same symmetry. Proposition 3.6. Each symmetry on a | | –graded geometry is involutive and thecenter of the symmetry is its isolated fixed point.Proof. The first part follows directly from the latter facts. The second part is ob-vious thanks to the fact that for each symmetry, its differential at the center actsas − id on the whole tangent space. (cid:3) YMMETRIES OF PARABOLIC GEOMETRIES 17 If we start with an effective geometry, we in addition have following consequencesof the latter Propositions: Proposition 3.7. Let ϕ and ψ cover two symmetries at x on some effective | | –graded geometry and suppose that their invariant Weyl structures coincide at x .Then ϕ = ψ over a neighborhood of x . Moreover, each covering of a symmetry isinvolutive.Proof. From the proof of Proposition 3.5 we get that ( ψ ◦ ϕ − ) ◦ σ ( v ) = σ ( v ) holds inthe case of effective geometries. The rest follows immediately from the formula. (cid:3) We can shortly summarize all these facts in the following way: • The symmetry can have several coverings, but all of them differ by the rightmultiplication by some element from the kernel and in the fiber over x , allof them share the same invariant Weyl structure at x . • Each symmetry with the center at x allows (in the fiber over x ) exactly oneinvariant Weyl structure at x . This does not depend on the choice of thecovering of the symmetry. • For effective geometries, each symmetry has exactly one covering.Let us illustrate the situation with symmetries on (non)effective geometries onthe homogeneous model of conformal geometry: Example . Conformal Riemannian structures. First, we have to show that homo-geneous models of conformal structures are symmetric. In Example 1.5 we startedwith G = O ( p + 1 , q + 1) and we would continue with this choice.We are looking for elements giving symmetries in the origin of this model, seeCorollary 1.10. For b = (cid:16) λ C 00 0 λ − (cid:17) ∈ G and V = (cid:16) X − X T J (cid:17) ∈ g − the adjointaction Ad b V is given by X λ − CX and we require λ − CX = − X . Thus welook for λ ∈ R \ { } and C ∈ O ( p, q ) such that CX = − λX for each X ∈ R p + q .Clearly, C has to be diagonal and all elements on the diagonal have to be equal to1 or − C is equal to 1 or − 1. We get two elements g = (cid:16) − E 00 0 − (cid:17) and g = (cid:16) − E 00 0 1 (cid:17) satisfying all conditions. Clearly, both elements belong to thegroup O ( p + 1 , q + 1). Then all elements inducing some symmetry in the origin areof the form (cid:16) − − Z ZJZ T E − JZ T − (cid:17) and (cid:18) Z − ZJZ T − E JZ T (cid:19) for all Z ∈ R p + q ∗ .The choice G = O ( p + 1 , q + 1) gives not an effective geometry. One can easilycompute that the kernel consists of elements (cid:16) E 00 0 1 (cid:17) and (cid:16) − − E 00 0 − (cid:17) . Thus theelements g and g differ by multiplication by some element from the kernel. Leftmultiplication by elements from the kernel induces identity on the base manifoldand then g and g give the same symmetry on the base manifold. Then eachpossible symmetry (in the origin) has exactly two possible coverings.It is possible to take an effective model, e.g. to start with P O ( p + 1 , q + 1), thefactor of the orthogonal group by its center. With this choice we clearly find exactlyone element from G satisfying all conditions – the class represented e.g. by theelement (cid:16) − E 00 0 − (cid:17) . Remark on relations to affine locally symmetric spaces. (1) Remark first,that there is a notion on geodesic symmetry for affine locally symmetric spaces. Each symmetry (locally) reverses geodesics going through its center. This propertydescribes the symmetry on some neighborhood of the center and can be used asa definition, see [11, 9]. Analogies of the affine geodesics for parabolic geometriesare generalized geodesics. They are defined as the projections of the flow lines ofa horizontal vector field. Detailed discussion on generalized geodesics can be foundin [8, 21].Our definition of the symmetry on | | –graded geometries implies that symme-tries are those automorphisms, which revert the ‘classes’ of generalized geodesics.More precisely, the symmetry at x maps each generalized geodesic going through x in some direction to some generalized geodesic going through x in the oppositedirection. This correspond to the fact that symmetries are not uniquely determinedin this case. We just saw this on homogeneous models, see Examples 1.11 and 3.8.There can exist a lot of different symmetries at one point on a | | –graded geometryand it is not reasonable to define symmetries only via reverting of geodesics.(2) On the affine (locally symmetric) space, there is exactly one normal coor-dinate system at the point (up to Gl ( n, R ) transformation) and one can use it todescribe nicely the symmetry at the point. In these coordinates the symmetry onlyreverts the straight lines going through the point.On parabolic geometries, there can exist many different normal coordinate sys-tems. They are given by the choice of the (second order) frame in the fiber. Weshowed that on | | –graded geometry carrying some symmetry at x , there is (up tosome transformation given by an element from G ) exactly one normal coordinatesystem σ u at x such that the (covering of the) symmetry only reverts the straightlines going through the point in these coordinates. Moreover, this fixes the connec-tion ∇ σ which is compatible with the symmetry and therefore, its geodesics (whichare generalized geodesics because the affine connection is normal) are reversed bythe symmetry.(3) Remark, that there is an equivalent definition of the symmetry on the mani-fold. One can define symmetry at x as a (locally defined) involutive automorphismsuch that the point x is the isolated fixed point of this automorphism, see [13] or[11]. Symmetries defined in this manner clearly satisfy the condition on the differ-ential and they are symmetries in our sense. Conversely, the Proposition 3.6 saysthat our symmetries correspond to the latter definition.(4) Finally remind the well known description of affine locally symmetric spaces.The pair ( M, ∇ ) is affine locally symmetric if and only if the torsion of ∇ vanishesand its curvature is covariantly constant. For symmetric | | –graded geometries, wehave just proved that the torsion of the Cartan connection (and thus of all Weylconnections) vanishes. In the next section, we will discuss the curvature.4. Further curvature restriction In the Theorem 2.5 we have indicated which geometries require some furtherstudy of the curvature and in this section, we deal only with them. We have todiscuss so called Weyl curvature, which can be defined for all parabolic geometriesvia Weyl structures (and which is related to the homogeneous component of thecurvature of homogeneous degree 2, see [7]). It is an analogy of the classical objectfrom conformal geometry, see [6] or [3, 4] for more detailed discussion on objectswhich generalize the classical conformal ones. YMMETRIES OF PARABOLIC GEOMETRIES 19 Finally we show some consequences for the concrete geometries. Let us first pointout that the motivation for this results is the article [13], where the author studiesthe projective case in the classical setup of affine connections. Our methods workfor all | | –graded parabolic geometries. Curvature restrictions. From our point of view, the most interesting | | –gradedgeometries are the ones, which allow some homogeneous component of curvatureof degree 2. Suppose there is some symmetry at each point of such geometry. Thenits curvature is of the form κ = κ : G → ∧ g ∗− ⊗ g because symmetric | | –graded geometries are torsion free. If we choose some Weyl structure σ , we get thedecomposition of σ ∗ κ = κ ◦ σ such that σ ∗ κ = σ ∗ κ + σ ∗ κ . The lowest homogeneitypart of the decomposition σ ∗ κ : G → ∧ g ∗− ⊗ g does not change, if we change the Weyl structure. This part is called Weyl curvature and is usually denoted by W .If ϕ is an automorphism of the parabolic geometry, then its curvature is invariantwith respect to ϕ . If we choose a Weyl structure σ such that ϕ ∗ σ = σ , then theWeyl curvature has to be invariant with respect to the underlying morphism. Westudy algebraic actions and covariant derivatives of the Weyl curvature W withrespect to Weyl connections. The existence of the invariant Weyl structure at thepoint is crucial for our considerations. Lemma 4.1. Let σ be an arbitrary Weyl structure on a | | –graded geometry andlet ϕ be a symmetry at x covered by some ϕ . Then { ξ, Υ } • W + 2 ∇ σξ W = 0 holds at x for all ξ ∈ X ( M ) , where Υ is defined by ϕ ∗ σ = σ + Υ .Proof. We take ∇ σξ W ( η, µ ) for each ξ, η, µ ∈ X ( M ) and we compute the pullbackof the connection with respect to the symmetry ϕ . At the point x we have( ϕ ∗ ∇ σ ) ξ W ( η, µ ) = ( T ϕ ⊗ T ϕ − ) . ∇ σT ϕ.ξ W ( T ϕ.η, T ϕ.µ ) = −∇ σξ W ( η, µ )for each ξ, η, µ ∈ X ( M ). Next, we have ϕ ∗ σ = σ + Υ for some Υ and this gives( ϕ ∗ ∇ σ ) ξ W ( η, µ ) = ∇ σ +Υ ξ W ( η, µ )for each ξ, η, µ ∈ X ( M ). If we put all together, we get that the identity −∇ σξ W ( η, µ ) = ∇ σ +Υ ξ W ( η, µ )holds at the point x . Using the formula for change of Weyl connection we canrewrite this as −∇ σξ W ( η, µ ) = ∇ σξ W ( η, µ ) + ( { ξ, Υ } • W )( η, µ ) . This holds for each ξ, η, µ ∈ X ( M ) at the point x and gives exactly the formula. (cid:3) As an easy consequence of the Lemma we get the following analogy of the resultfrom affine locally symmetric spaces: Theorem 4.2. Suppose there is a symmetry with the center at x on a | | –gradedgeometry. Then there exists a Weyl connection ∇ σ such that ∇ σ W = 0 at thepoint x . The connection corresponds to the invariant Weyl structure at x . Proof. Let ϕ cover a symmetry ϕ with the center at x and let σ be the Weylstructure invariant with respect to ϕ at x . Thus ϕ ∗ σ = σ holds at the point x . Onecan see all from the Lemma 4.1. In this case, we have Υ = 0 at x and the algebraicbracket from the expression in the Lemma has to vanish for all ξ ∈ X ( M ) at x .Then 2 ∇ σξ W = 0 holds for all ξ and this implies ∇ σ W = 0 at x . (cid:3) In some sense, this is analogous to the classical results. On the affine locallysymmetric space, there is exactly one connection which is invariant with respect toall symmetries and we know that its curvature is covariantly constant with respectto the connection.In the case of symmetric | | -graded geometries, there is the class of Weyl con-nections and we showed that at each point, there is at least one of them such thatthe Weyl curvature is covariantly constant at the point. Algebraic restrictions. Using all latter facts we show some algebraic restrictionon the Weyl curvature of symmetric | | –graded geometries. Let us first introducesome conventions.Let ϕ and ψ cover two different symmetries ϕ and ψ with the center at x ona | | –graded geometry. We showed above that in the fiber over x , the symmetrieshave different invariant Weyl structures at x . Let σ be ϕ -invariant Weyl structureat x , i.e. ϕ ∗ σ = σ at x . Then σ cannot be ψ –invariant at x and we have ψ ∗ σ = σ +Υwhere Υ is nonzero at x . In this way, we can find such Υ for each two coverings ϕ and ψ of two (different) symmetries. In fact, this Υ does not depend on the choiceof the coverings of ϕ and ψ at the point x because invariant Weyl structures of twodifferent coverings of the same symmetry have to coincide along the fiber over x . Inthe sequel, we call the form Υ at x the difference between ϕ and ψ . In the future,we often use the fact that this difference is nonzero at x , but we do not need theexact value. From this point of view, it is not important which symmetry is thefirst one and which is the second one. Proposition 4.3. Assume there are two different symmetries at the point x on a | | –graded geometry. Then { ξ, Υ } • W = 0(1) holds at x for any ξ ∈ X ( M ) , where Υ is the difference between the symmetries.Proof. Let ϕ and ψ be two different symmetries at x with coverings ϕ and ψ andlet Υ be the difference between ϕ and ψ , i.e. ϕ ∗ σ = σ and ψ ∗ σ = σ + Υ hold atthe center x for some Weyl structure σ and Υ is nonzero at x . The Lemma 4.1 andTheorem 4.2 give that ∇ σξ W = 0 , { ξ, Υ } • W + 2 ∇ σξ W = 0hold for all ξ ∈ X ( M ). If we put this together, we get the required expression. (cid:3) Corollary 4.4. If there are two different symmetries with the center at x on a | | –graded geometry then {{ ξ, Υ } , W ( η, µ )( ν ) } − W ( {{ ξ, Υ } , η } , µ )( ν ) − W ( η, {{ ξ, Υ } , µ } )( ν ) − W ( η, µ )( {{ ξ, Υ } , ν } ) = 0(2) holds at x for all ξ, η, µ, ν ∈ X ( M ) , where Υ is the difference between the symme-tries. YMMETRIES OF PARABOLIC GEOMETRIES 21 Proof. The expression { ξ, Υ }• W is of the type ∧ T ∗ M ⊗ T ∗ M ⊗ T M for any vectorfield ξ and we evaluate it on vector fields η, µ and ν . We get:( { ξ, Υ } • W )( η, µ )( ν ) = {{ ξ, Υ } , W ( η, µ )( ν ) } − W ( {{ ξ, η } , η } , µ )( ν ) − W ( η, {{ ξ, Υ } , µ } )( ν ) − W ( η, µ )( {{ ξ, Υ } , ν } ) . This implies directly the required formula. (cid:3) The latter Proposition gives some restriction on the curvature, which is not tooclear. But we can use the formula (1) to find some more clear restriction on thecurvature for the interesting geometries. We should discuss the action of variouselements, which can be written as the algebraic bracket of the difference and somevector field. We would like to find some elements which act on the Weyl curvaturein a sufficiently simple way. Then we could understand better the consequences ofthe restriction. Theorem 4.5. Suppose there are two different symmetries at x on a | | –gradedgeometry. Let Υ = [[ u, Z ]] be their difference at x for some Z ∈ g and u from thefiber over x and suppose there exists X ∈ g − such that [ X, Z ] acts by the adjointaction diagonalizable on g − with eigenvalues a , . . . , a k such that a i + a i + a i − a i = 0 for arbitrary choice of them. Then the Weyl curvature vanishes at x andthus the whole curvature vanishes at x .Proof. Remind that Υ has to be nonzero at x because the symmetries are different.For various vector fields ξ , we would like to discuss the action of the elements of theform { Υ , ξ } on the Weyl curvature, see Proposition 4.3. First possible simplificationgives us the formula (2) which reduces the discussion of the action on Weyl curvatureto the action on the tangent vectors. We will discuss the case when the suitableelements act simply by multiplication by numbers.Let us now describe the situation using the definition of tractor bundles andreduce it to the computation in coordinates, see Section 1. Let u be the frame fromthe fiber over x such that Υ( x ) = [[ u, Z ]] for Z ∈ g . For each vector field ξ wecan then write ξ ( x ) = [[ u, X ]] for some X ∈ g − . In this way, we have { ξ, Υ } ( x ) =[[ u, [ X, Z ]]] and so on.We would like to discuss the action of the latter bracket on vector fields at x . Incoordinates at the frame u , we simply study the adjoint action of [ X, Z ] on the whole g − . Let X ∈ g − be as in the assumption. Thus [ X, Z ] acts by the adjoint actiondiagonalizable on g − with eigenvalues a , . . . , a k and g − decomposes into theeingenspaces. Then the tangent space at x decomposes in the same way by the actionby { ξ, Υ } where ξ ( x ) = [[ u, X ]]. We have found ξ ∈ X ( M ) giving ‘understandable’action of the bracket.Now, the formula (2) says how to understand the action of this element on theWeyl curvature. The Weyl curvature lives in the component of ∧ T ∗ M ⊗ T ∗ M ⊗ T M and decomposes with respect to the decomposition of the tangent space. Thusup to the choice of the frame u , the action of the bracket on each component ismultiplication by the sum of suitable eigenvalues. If this sum is always nonzero,then the Weyl curvature has to vanish.If there is some symmetry with the center at x , then the torsion vanishes at x .Existence of two different symmetries at x satisfying the condition kills the Weylcurvature at x and then κ = κ . But κ has to vanish too, the reason is the sameas in the case of the torsion. (cid:3) Now, we can simply verify the latter condition for each geometry separately. Wewill see later that the ‘simplification’ given by the choice of concrete ξ is sufficient toget quite strong restrictions in concrete geometries. We will discuss conformal andprojective structures in detail. The almost Grassmannian and almost quaternionicstructures behave analogously and we give here only short remark rather than theprecise description. Conformal structures. The Theorem 4.5 reduces the discussion of the Weyl cur-vature of geometries with more than one symmetry at the point to simple algebraiccondition on Lie algebras of the corresponding geometry. We discuss here, whetherthe condition can be satisfied in the case of conformal geometry. The computationswill be performed in the setting of Example 1.5. Theorem 4.6. Suppose there are two different symmetries with the center at x on the conformal geometry of arbitrary signature and denote Υ their difference.Suppose that this Υ has nonzero length at x . Then the curvature κ vanishes at x .Proof. We will discuss the condition from the Theorem 4.5. Up to the choice of theframe, the difference can be represented by some matrix Z = (cid:16) V 00 0 − JV T (cid:17) ∈ g forsome nonzero V ∈ R p + q . We choose X = (cid:16) JV T − V (cid:17) ∈ g , just the dual of Z .The Lie bracket of the elements is h(cid:16) JV T − V (cid:17) , (cid:16) V 00 0 − JV T (cid:17)i = (cid:18) − V JV T V JV T (cid:19) . Here V JV T correspond to the square of the length of Υ and it is nonzero if andonly if Υ has nonzero length. Under this condition, the latter element clearly actsas a multiplication. More precisely, we have " − V JV T V JV T ! , Y − JY T ! = V JV T Y − V JV T JY T ! . Thus the whole g − is one eigenspace and the condition on the sum of eigenvaluesis always satisfied. Then the curvatures has to vanish, see Theorem 4.5. (cid:3) Remark . One can also reduce some computation by the observation, that thebracket is just multiple of the grading element in the conformal case. Remind, thatthe grading element is the only element E ∈ g with the property [ E, Y ] = jY forall Y ∈ g j . It exists for each parabolic geometry and one can verify that in theconformal geometry, it is of the form (cid:16) − (cid:17) . We just multiply it by the number − V JV T = −| V | . Corollary 4.8. Suppose there are two different symmetries with the center at x onthe conformal geometry of positive definite or negative definite signature. Then thecurvature vanishes at x .Proof. In these cases, the length of a nonzero vector is always nonzero. The restfollows immediately. (cid:3) YMMETRIES OF PARABOLIC GEOMETRIES 23 Projective geometries. Let us make now similar discussion for projective ge-ometries. We will study the vanishing of the curvature at the point with more thanone symmetry. We are again interested in the condition in Theorem 4.5. We usethe notation introduced in Example 1.11. Theorem 4.9. Suppose there are two different symmetries with the center at x ona projective geometry. Then the curvature κ of the geometry vanishes at x .Proof. We will discuss the condition in the Theorem 4.5. In contrast to the con-formal case, we cannot find the bracket in the form of a multiple of the gradingelement. One can find the grading element and compute the bracket of g − and g in general. Up to the choice of the frame, we represent the difference by matrix Z = ( V ) where V = (1 , , . . . , ∈ R m ∗ . (We take concrete matrix, because incontrast to the conformal case, the computation in general is not transparent.) Wechoose X = (cid:0) V T (cid:1) . The Lie bracket of the elements is h(cid:16) V T (cid:17) , (cid:16) V (cid:17)i = (cid:16) − V V T V T V (cid:17) . For our choice, V V T = 1 and matrix V T V has 1 on the first row and column andzero elswere. The action of the element on g − is following: − ... 00 1 0 ... 00 0 0 ... ... ... ... ... ... ... , ... a ... b ... ... ... ... ... ... c ... = ... a ... b ... ... ... ... ... ... c ... . Thus g − decomposes into the eigenspaces for eigenvalues 1 and 2. The conditionon the sum of eigenvalues is always satisfied and we have no other restriction. Thenthe curvatures has to vanish, see Theorem 4.5. (cid:3) Corollary 4.10. For projective geometries, there can exist at most one symmetryat the point with nonzero curvature. If there are two different symmetries at eachpoint, then the geometry is locally flat. Remark on further geometries. We just shortly summarize analogous resultsfor the remaining two geometries. Theorem 4.11. (1) Suppose there are two different symmetries with the centerat x on the almost Grassmannian geometry of type (2 , q ) or ( p, and denote Υ their difference. Suppose that this Υ has maximal rank at x . Then the curvature κ vanishes at x .(2) Suppose there are two different symmetries with the center at x on an almostquaternionic geometry. Then the curvature κ of the geometry vanishes at x . The proofs are similar to the projective case and can be found in [19]. The moreprecise discussion and examples of the almost Grassmannian case can be also foundin [20]. Remark . The condition of maximal rank for the almost Grassmannian geome-tries of the latter type means that the rank of the difference is 2 at x . Remind thatfor the almost Grassmannian geometries, the other cases ( p > q > 2) are notinteresting, because each symmetric geometry of this type has to be locally flat.Let us also note that in the lowest dimension ( p = q = 2), the almost Grass-mannian geometry correspond to the conformal geometry of indefinite type and thecondition on the length of the Υ agree with the condition on its rank. References [1] D.V. Alekseevski˘ı, Groups of conformal transformations of Riemannian spaces . (Russian)Mat. Sb. (N.S.) 89(131) (1972), 280–296.[2] A. ˇCap, Two constructions with parabolic geometries , Proceedings of the 25th Winter Schoolon Geometry and Physics, Rend. Circ. Mat. Palermo (2) Suppl. No. 79 (2006), 11–37.[3] A. ˇCap, R. Gover, Tractor Bundles for Irreducible Parabolic Geometries , SMF, S´eminaireset Congr`es, n.4, (2000), 129-154.[4] A. ˇCap, R. Gover, Tractor Calculi for Parabolic Geometries , Trans. Amer. Math. Soc. 354(2002), 1511-1548.[5] A. ˇCap, H. Schichl, Parabolic geometries and canonical Cartan connection , Hokkaido Math.J. 29 (2000), 453–505.[6] A. ˇCap, J. Slov´ak, Weyl Structures for Parabolic Geometries , Math. Scand. 93, (2003), 53–90.[7] A. ˇCap, J. Slov´ak, Parabolic Geometries, Mathematical Surveys and Monographs, AMS Pub-lishing House, to appear[8] A. ˇCap, J. Slov´ak, V. ˇZ´adn´ık, On Distinguished Curves in Parabolic Geometries , Transfor-mation Groups, 9, (2004) 143-166.[9] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, Inc.,1978, 628pp.[10] W. Kaup, D. Zaitsev, On symmetric Cauchy-Riemann manifolds , Adv. Math. 149 (2000),145–181.[11] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry. Vol II, John Wiley & Sons,New York-London-Sydney, 1969, 470pp.[12] I. Kol´aˇr, P.W. Michor, J. Slov´ak, Natural Operations in Differential Geometry, Springer-Verlag, 1993, 434pp.[13] F. Podesta, A Class of Symmetric Spaces , Bulletin de la S.M.F., tome 117, n.3, 1989, p.343-360.[14] R.W. Sharpe, Differential geometry: Cartan’s generalization of Klein’s Erlangen program,Graduate Texts in Mathematics 166, Springer-Verlag 1997.[15] J. ˇSilhan, Algorithmic computations of Lie algebra cohomologies , Proceedings of the WinterSchool on Geometry and Physics, Srn´ı 2002, Suppl. Rendiconti Circolo Mat. Palermo, SerieII, 2003, 7p. 191-197.[16] J. ˇSilhan, A real analog of Kostant’s version of the Bott-Borel-Weil theorem , J. Lie Theory14 (2004), no. 2, 481–499.[17] K. Yamaguchi, Differential systems associated with simple graded Lie algebras , AdvancedStudies in Pure Mathematics 22 (1993), 413–494.[18] L. Zalabov´a, Remarks on Symmetries of Parabolic Geomeries , Arch. Math., 42 Suppl., ‘Pro-ceedings of the Winter School Geometry and Physics 2006’, 357-368.[19] L. Zalabov´a, Symmetries of almost Grassmannian geometries , Proceedings of DifferentialGeometry and Its Applications, Olomouc, 2007.[20] L. Zalabov´a, V. ˇZ´adn´ık, Remarks on Grassmannian Symmetric Spaces , to appear in Arch.Math.[21] V. ˇZ´adn´ık, Generalised Geodesics , Ph.D. thesis, 2003, 65 pp. Masaryk University, Brno, Czech Republic and International Erwin Schr¨odingerInstitute for Mathematical Physics, Vienna, Austria E-mail address ::