Essential Killing fields of parabolic geometries
We study vector fields generating a local flow by automorphisms of a parabolic geometry with higher order fixed points. We develop general tools extending the techniques of , , and , and we apply them to almost Grassmannian, almost quaternionic, and contact parabolic geometries, including CR structures. We obtain descriptions of the possible dynamics of such flows near the fixed point and strong restrictions on the curvature; in some cases, we can show vanishing of the curvature on a nonempty open set. Deriving consequences for a specific geometry entails evaluating purely algebraic and representation-theoretic criteria in the model homogeneous space.
Dedicated to Michael Eastwood on the occasion of his 60th birthday.
An array of results in differential geometry tell us that geometric structures admitting a large group of automorphisms are special and must have a particularly simple form. For example, a Riemannian manifold with of maximum possible dimension must have constant sectional curvature and thus be a space form. More generally, the maximal dimension for the Lie algebra of Killing vector fields on a Riemannian manifold, or for the Lie algebra of infinitesimal automorphisms for many classical geometric structures, can be only attained on open subsets of a homogeneous model.
In some cases, the existence of a single automorphism or infinitesimal automorphism of special type restricts the geometry. Special automorphisms that exists for some geometric structures are those that equal the identity to first order at a point; note that because of the exponential map, Riemannian metrics never admit such automorphisms, except the identity. The projective transformations of projective space , on the other hand, do include such automorphisms: there is with and . The space viewed as a homogeneous space of the group of projective transformations is the model for classical projective structures. Such a structure on a manifold is an equivalence class of torsion-free linear connections on having the same sets of geodesics up to reparametrization. An automorphism is a diffeomorphism of preserving , or, equivalently, preserving the corresponding family of geodesic paths as unparametrized curves.
For connected, an automorphism of a classical projective structure on is uniquely determined by its two–jet at a single point. Non–trivial automorphisms fixing a point to first order are examples of essential automorphisms—ones not preserving any connection in the projective class . Nagano and Ochiai  proved that if a compact, connected manifold with a torsion-free connection admits a nontrivial vector field for which the flow is projective and trivial to first order at a point , then is projectively flat on a neighborhood of —that is, locally projectively equivalent to .
Pseudo–Riemannian conformal structures may also admit non–trivial automorphisms which equal the identity to first order in a point. In this case, Frances and the second author prove analogous results in  and . Their theorems say that if a conformal vector field vanishes at a point , and if the flow is unbounded but has precompact differential at , then the manifold is conformally flat on a nonempty open set with .
Both proofs make use of the Cartan geometry canonically associated to the structures in question and of the contracting—though not necessarily uniformly contracting—dynamics of the given flows. In both cases, the Cartan geometry is a parabolic geometry, one for which the homogeneous model is for G a semisimple Lie group and a parabolic subgroup. An introduction to the general theory of parabolic geometries can be found in . See  and  for general results on automorphisms and infinitesimal automorphisms.
In this article, we develop machinery to apply these ideas to study the behavior of a certain class of flows fixing a point that include the projective and conformal flows described above, in the general setting of parabolic geometries. Our results lead to descriptions of the possible dynamics of such flows near the fixed point and to strong restrictions on the curvature. In some cases, we can show vanishing of the curvature on a nonempty open set. Deriving consequences for a specific geometry entails evaluating purely algebraic and representation-theoretic criteria for the pair .
1.1.1. Cartan geometries of parabolic type
Let be a semisimple Lie group with Lie algebra . A parabolic subalgebra of can be specified by a –grading for some positive integer , which is a grading of of the form such that no simple ideal is contained in the subalgebra , and such that the subalgebra is generated by . The parabolic subalgebra determined by the grading is then , and a parabolic subgroup is a subgroup with Lie algebra . It is a fact that
where is the normalizer and is its connected component of the identity. The center of contains the grading element , for which each is an eigenspace of with eigenvalue . The –gradings of a given Lie algebra correspond to subsets of the simple roots when is complex, and subsets of simple restricted roots for real, associated to a choice of Cartan subalgebra (which is maximally non–compact in the real case); see section 3.2 of .
Defining makes into a filtered Lie algebra such that . The parabolic subgroup acts by filtration–preserving automorphisms under the adjoint action. The subgroup preserving the grading of has Lie algebra . Denote , and let be the corresponding subgroup; it is unipotent and normal in , and is a diffeomorphism. Then , and it is closed and reductive.
Let be a Lie group with Lie algebra and a closed subgroup. A Cartan geometry on modeled on the pair is a triple , where
is a principal -bundle
is the Cartan connection, satisfying
for all , the restriction is a linear isomorphism
for all , the pullback
for all , if is the fundamental vector field , then .
The Cartan connection generalizes the left-invariant Maurer-Cartan form on . The following curvature is a complete obstruction to local isomorphism of to the homogeneous model .
The curvature of the Cartan connection is the two–form given by
A parabolic geometry on a manifold is a Cartan geometry on modeled on for a semisimple Lie group and a parabolic subgroup.
The Cartan connection gives rise to natural local charts on as follows. To each corresponds an –constant vector field , characterized by . Note that is the fundamental vector field if . For any , and sufficiently small , define to be the image of under the time-one flow along . There is a neighborhood of on which is defined and a diffeomorphism onto an open subset of . Composing the projection with the restriction of to any linear subspace in complementary to , we obtain a local chart on . The exponential map gives rise to a notion of distinguished curves and to normal coordinates on a parabolic geometry:
Consider a parabolic geometry on of type .
For , an exponential curve in is the projection to of a curve for some . It is a distinguished curve of the geometry if .
A distinguished chart on is a chart with values in obtained as a local inverse of .
For a given parabolic model, exponential and distinguished curves can be classified by the geometric type of the initial direction (see section 1.2 below). Section 5.3 of  contains thorough descriptions of the classes of distinguished curves for many parabolic models. In the next section, we introduce normal charts, with values in tangent spaces.
1.1.2. Adjoint tractor bundle and infinitesimal automorphisms
Natural vector bundles on parabolic geometries modeled on can be obtained as associated bundles to the Cartan bundle. Given a representation of , form . Using the Cartan connection, such bundles can sometimes be identified with tensor bundles; for example, the adjoint representation restricted to descends to the quotient vector space , and . The main idea to define this isomorphism is to note that for each with , the linear isomorphism induces a linear isomorphism .
For with , the linear isomorphism composed with the normal coordinate chart gives a diffeomorphism from an open neighborhood of in onto an open neighborhood of , which we will refer to as a normal coordinate chart for . By construction, a straight line through corresponds to a distinguished curve through with initial direction .
Two further associated bundles to the Cartan bundle will play a role in the sequel. The Killing form of gives an identification of with , so . The adjoint tractor bundle is and is useful for studying infinitesimal automorphisms. The filtration of gives rise to a filtration by smooth subbundles. Since and , we see that and . Denote by the resulting natural projection.
The curvature of from definition 1.2 can be naturally viewed as an element . Indeed, from the defining properties of it follows easily that is horizontal and –equivariant and thus corresponds to a form as above.
Vector fields on are in bijective correspondence with –valued smooth functions via . Equivariance of immediately implies that is a –equivariant function if and only if is a right––invariant vector field. The space of smooth sections of can be naturally identified with the space of –invariant vector fields on ; note that these descend to . On the bundle , the corresponding projection is .
An automorphism of is a principal bundle automorphism that preserves . These form a Lie group, which will be denoted . An infinitesimal automorphism is given by such that , where denotes the Lie derivative. (For a pseudo-Riemannian metric, infinitesimal automorphisms are called Killing fields.) An infinitesimal automorphism descends to a vector field on . The resulting subalgebra of will be denoted below; note these vector fields are not assumed to be complete.
1.1.3. Normal parabolic geometries and harmonic curvature
Parabolic geometries encode certain underlying geometric structures. First note that via the isomorphism , a parabolic geometry of type gives rise to a filtration of the tangent bundle, where . This filtration gives rise to a filtration of by homogeneity: is called homogeneous of degree if for and the value .
The geometry is regular if the curvature 2-form is homogeneous of degree at least 1. A Cartan geometry is torsion-free if has values in ; torsion-free implies regular. Now assuming regularity, the underlying structure of a parabolic geometry consists of the filtration of the tangent bundle and a reduction of structure group of the associated graded of this filtered bundle to . Conversely, any such structure of a filtration with a -reduction can be obtained from some regular parabolic geometry.
These geometric structures are equivalent, in the categorical sense, to regular parabolic geometries satisfying an additional condition on called normality. The Lie algebra homology differentials for the Lie algebra with coefficients in the module defines a –equivariant homomorphism , which is traditionally denoted by and called the Kostant codifferential. For , this homomorphism gives on the level of associated bundles a natural bundle map , also denoted by . Now the geometry is called normal if .
The projective and conformal structures mentioned above correspond to –gradings. Other geometric structures arising from parabolic Cartan geometries include almost-Grassmannian and almost-quaternionic structures, hypersurface–type CR structures, path geometries, and several types of generic distributions.
The equivalence in the categorical sense implies that any automorphism of the underlying structure uniquely lifts to an automorphism of the parabolic geometry. The analogous result for vector fields says any infinitesimal automorphism of the underlying structure lifts uniquely to such that . Conversely, projecting an infinitesimal automorphism of to gives an infinitesimal automorphism of the underlying structure.
The normality condition for parabolic geometries can also be used to extract the essential part of the curvature of the canonical Cartan connection. As mentioned above, the Kostant codifferential induces bundle maps
Since these maps come from a homology differential, the composition of the two bundle maps above is zero, so there are natural subbundles . By construction, the quotient bunde can be realized as , and the latter Lie algebra homology group can can identified with the Lie algebra cohomology group . For a regular normal parabolic geometry, the curvature actually is a section of . The harmonic curvature is the image of under the obvious quotient projection.
General theorems assert that no information is lost in passing from to . First, vanishing of on an open subset implies vanishing of on . In fact, there is a natural differential operator such that . The crucial advantage of the harmonic curvature is that one can show that the representation is always completely reducible, so the corresponding associated bundle is a simpler geometric object than . The structure of can be computed with Kostant’s version of the Bott–Borel–Weil theorem—see section 3.3 of . We will use without further citation the resulting descriptions of harmonic curvature components for the individual geometries we discuss.
1.2. Higher order fixed points and the main questions
Let be a regular normal parabolic geometry of type . Let , and denote the induced vector field on by and by the corresponding section of . For , we have if and only if , equivalently if for any .
The infinitesimal automorphism , or the corresponding section of , has a higher order fixed point at if . In this case, the isotropy at of (or of ) is the element of the cotangent space corresponding to .
Via the isotropy corresponds to an element in . The geometric type of the infinitesimal automorphism at is defined to be the –orbit of this element, which is independent of the choice of . These geometric types give rise to an initial classification of higher-order fixed points. For example, in the conformal case, there is a natural inner product on up to scale, so isotropies can be postive, null, or negative. For algebraically more complicated models, as for CR structures, the cotangent bundle has a natural filtration induced by , which leads to a variety of possible geometric types of isotropies. These will be discussed in detail in several examples below. In all cases we consider there are only finitely many orbits, and in general there are finiteness results of E. Vinberg .
Main Questions: What special dynamical properties are shared by infinitesimal automorphisms admitting a higher order fixed point with isotropy of a certain geometric type? What curvature restrictions are implied by existence of such an automorphism? Which types of higher order fixed points in imply that is locally flat on a nonempty open set with ?
The concept of essential automorphisms, which previously existed for conformal and projective structures, has recently been extended to all parabolic geometries by J. Alt in . It is immediate from his proposition 3.4 that a Killing field with a higher order fixed point is essential.
We develop general tools in section 2 that give the precise action of a flow on specific curves emanating from a higher order fixed point. Several of these propositions generalize the tools of  and . We can apply them to recover the previously cited theorems on higher order fixed points for projective and conformal flows. These applications are presented in a separate article .
In section 3.1, we apply these tools to -almost-Grassmannian structures to describe the two types of higher order fixed points in these geometries, and we show in theorem 3.1 that if the geometry is torsion-free, then existence of either type implies flatness on a nonempty open set. The proofs in this section easily adapt to prove an analogous result for almost-quaternionic structures, without any torsion-freeness assumption (see theorem 3.7).
Next we prove a general result for parabolic contact structures (see section 3.3): another natural generalization of the hypotheses in the projective and conformal results  and  is to assume that a flow fixes a point and has trivial derivative at , which in the parabolic contact case is a stronger assumption than being a higher order fixed point. Under this hypothesis, we prove in theorem 3.9 that the curvature vanishes on an open set with in its closure; in some cases, we can further deduce flatness on a neighborhood of .
Finally, in section 3.4, we treat partially integrable almost-CR structures. Theorem 3.11 says the harmonic curvature always vanishes at a higher order fixed point, and certain types of higher order fixed points imply flatness on a nonempty open set. A consequence of this theorem is a local version of the Schoen-Webster theorem on automorphisms of strictly pseudoconvex CR structures (see theorem 3.16; compare also  and .). For general nondegenerate CR structures, we are left with the following question. (Beloshapka  and Loboda  have proved the answer is no in the case of real-analytic hypersurfaces of complex manifolds.)
Is there a non-flat partially integrable almost-CR manifold, the automorphism group of which has a higher-order fixed point?
2. General results
We begin with a basic proposition from  that computes the holonomy of an automorphism of a Cartan geometry with isotropy in terms of the action of on .
[9, prop 4.3] Let . Suppose
for some and
is defined for in an interval around and for some
in , where with , and is a diffeomorphism fixing .
Then the corresponding equation holds in : is also defined on , and
The proposition says that in normal coordinates for centered at the fixed point , the automorphism resembles the model automorphism acting on with fixed point . The proposition is related to the comparison maps studied in the recent paper .
Before stating our first general results on higher order fixed points, we introduce some terminology.
The strongly fixed set of a given geometric type of the flow generated by is the set of all higher order fixed points of that type. For a higher order fixed point , the term strongly fixed set will mean the strongly fixed set of the type of .
Given a neighborhood of a higher order fixed point the strongly fixed component of in is the set of all points that can be reached from by a smooth curve contained in the intersection of with the strongly fixed set of . The higher order fixed point is called smoothly isolated if equals the strongly fixed component in some neighborhood.
For any and any choice of , the Cartan connection gives identifications and such that the duality between the two spaces is induced by the Killing form of . Then any corresponds to an element . Put
A different choice is of the form for some , which leads to ; ; . By point 2(b) in definition 1.1 the images of these subsets in determine subsets which are independent of the choice of . Of course, is a linear subspace.
The subset and the subspace determined by as above are called, respectively, the normalizing subset and the commutant of in .
The following is a consequence of proposition 2.1 for the local behavior of an infinitesimal automorphism around a higher order fixed point.
Consider a Cartan geometry modeled on and with higher order fixed point with isotropy .
For any , there is an exponential curve emanating from in the direction consisting of fixed points for . If , then this curve lies in the strongly fixed set of .
If , then there is a –dimensional submanifold through contained in the strongly fixed set of , with .
For some neighborhood of any point in the strongly fixed component of in can be reached by an exponential curve emanating from in a direction belonging to . In particular, if , then is smoothly isolated.
Proof: Choose and set . Let be a neighborhood of in on which the restriction of is defined and is a diffeomorphism onto its image. For , there is an element such that . Let , defined for small .
In we have and
which lies in since . If , then can be chosen in , and then . It follows from proposition 2.1 that in , for all sufficiently small , , and ,
Now (1) follows for because the rightmost term is in . For , differentiate with respect to at time 0 to see that coincides with the fundamental vector field generated by for all .
For point (2), note that we can choose a –dimensional subspace with . Extend to a linear subspace of which is complementary to . Then can be restricted to an open neighborhood of zero to obtain a submanifold chart as required.
For (3), let be a smooth curve emanating from which lies in the strongly fixed set of . Then there is a lift of , which we may assume begins at , satisfying
for some with corresponding fundamental vector field ; moreover, each is conjugate in to , and . Now the –conjugacy class of is in bijection with , where is the stabilizer in of under the adjoint action. A smooth path in this quotient can be lifted to . Hence on a sufficiently small interval, we obtain a smooth path through the identity in such that . Then let . For this lift,
Therefore for sufficiently small . Fix for which , so there is such that . Then
for all . On the other hand, the expression above also equals
Therefore , so for any sufficiently small that , we have , and thus , which completes the proof of (3).
In most cases we discuss, we can strengthen this result by showing that the curves in (1) are distinguished curves for the geometry and describing the submanifold in (2) in terms of normal coordinates. These improvements will be presented at the end of section 2.
To proceed further, we need an analog of the concept of holonomy sequences associated to sequences of automorphisms of a Cartan geometry, which appeared in . The notion was further developed in later papers, including , , , and . The following definition for flows is the most useful variation for our purposes.
Let be a flow by automorphisms of , and let . A path is a holonomy path at with attractor for if there exists a path and a point in with
The following proposition provides a condition under which a holonomy path at one point can be propagated to nearby points.
Let be a flow by automorphisms of . Let be a holonomy path at with attractor and path as in definition 2.7. Suppose that for some , is defined for all and as . Then is a holonomy path at with attractor .
Proof: We have
Then , and
A holonomy path at leads to restrictions on the possible –invariant sections of any bundle associated to the Cartan bundle. Given a representation of , a principal bundle automorphism of gives rise to an automorphism of the associated bundle . If also preserves , and the associated bundle is a tensor bundle, then this automorphism is the one functorially associated to . Smooth sections of correspond to smooth -equivariant maps —that is, for any . The pullback of a section by corresponds to the precomposition of with . In particular, if corresponds to a –invariant section for a flow, then for all .
The Cartan curvature , and components of the harmonic curvature for parabolic geometries, are both invariant under automorphisms because the corresponding sections are constructed naturally from the Cartan connection. For the Cartan curvature, the representation is , which in general is rather complicated; for the components of the harmonic curvature, in contrast, the representation is always irreducible. The number of harmonic curvature components and the form of the corresponding representations varies according to the type of geometry in question.
Let be a holonomy path for corresponding to with attractor , and let be the equivariant function corresponding to a –invariant section of . Put as in definition 2.7.
Then as .
Assume moreover that is contained in a –parameter subgroup of that is diagonalizable on , and let be an eigenspace decomposition with eigenvalues given by functions , . Let be the component of in .
If as , then .
If , then , where is any norm on ; in particular, if but does not tend to as , then .
Recall that for functions and defined on , the notation means there exist nonzero constants such that
For (1), we see that the -equivariance of implies
and -invariance implies ; now (1) follows.
For (2) we have by (1) that
Since for , the first property follows immediately. For the second, also use that .
Let be a holonomy path at and a -representation satisfying the hypotheses of part (2) of proposition 2.9 above. Let and , , be as above.
The stable subspace for , denoted , is the sum of the eigenspaces for which is bounded as .
The strongly stable subspace for , denoted , is the sum of eigenspaces with as .
Part (2) of proposition 2.9 says that we must always have , and if , then also .
A crucial fact for the sequel is that in the case of parabolic geometries, nice holonomy paths can be obtained from purely algebraic data. Recall that an –triple in a Lie algebra is formed by elements such that , , and . The Jacobson–Morozov theorem says that any element in the semisimple Lie algebra which is nilpotent in the adjoint representation can be completed to an –triple: there exists such that , , and form an –triple.
Let and . As above, a choice of associates to an element , which is nilpotent. We define to be the non–empty set of all , such that , , and form an –triple. As for the normalizing set and the commutant, we obtain a subset which is independent of the choice of .
For , the non–empty subset defined above is called the counterpart set of .
Now we can precisely compute the action of on certain curves from in any direction belonging to the counterpart set of the isotropy.
Let be modeled on , and let have a higher order fixed point at with isotropy . Let , and let be the elements associated to and , respectively, for a choice of ; set .
Then there exists an exponential curve , for some , such that , , and
Moreover, for each such , there is such that is a holonomy path at with attractor .
Proof: In , compute
The –triple in formed by , , and gives rise to a Lie algebra homomorphism , which locally integrates to a group homomorphism . The above equation then shows that in
Proposition 2.1 shows that the local flow of satisfies
for sufficiently small . Fixing such a value for , the path evidently satsfies , and . Define to be this latter curve, valid on some nonzero interval .
Note , and for all . Finally, equation (1) says that , which converges to for as .
Basic representation theory says that in a standard basis of , the semisimple element acts diagonalizably with integer eigenvalues in any finite-dimensional complex representation. Thus for an –triple , , in , the endomorphism is diagonalizable on , so is a semisimple element. Further, acts diagonalizably with integer eigenvalues on any finite-dimensional complex representation of . We will assume all representations are finite-dimensional below.
If we assume in addition that , then , so acts diagonalizably on . Now is reductive, so it is the direct sum of the center plus a semisimple subalgebra. The component of in the center acts trivially under ad, so the semisimple component of acts diagonalizably under ad, and hence in any finite-dimensional representation. Therefore, acts diagonalizably in any finite-dimensional representation of in which the center acts diagonalizably. All representations corresponding to the components of the harmonic curvature have this property because they are subquotients of representations of in which the center of is contained in a Cartan subalgebra. (All real representations of interest we are aware of have this property.)
Let , and let be a representation of on which acts diagonalizably with eigenspace decomposition and eigenvalues , .
The stable subspace for , denoted , is the sum of the eigenspaces for which .
The strongly stable subspace for , denoted , is the sum of eigenspaces with .
Now our results have the following useful formulation:
In the setting of proposition 2.12, assume that and let be a representation of . Let .
If , then any –invariant section of vanishes on .
If , then any –invariant section of which vanishes at vanishes on all of .
Suppose that and that all eigenvalues of on are non–positive with the -eigenspace equal to . Then any –invariant section of that vanishes at each fixed point of the same geometric type as in a neighborhood also vanishes on an open neighborhood of and thus on an open subset containing in its closure.
Proof: The –eigenspace for is the eigenspace for with eigenvalue . Thus and for the holonomy path . After definition 2.10, we observed that if is the function corresponding to an invariant section of the bundle , then , and if , then . These facts together with proposition 2.12 yield (1) and (2).
The additional assumption in (3) implies that for any , the limit of as exists, and equals some . Taking a point over a point close enough to , we can thus invoke proposition 2.8 to say that is a holonomy path at with attractor (assuming sufficiently close to ). But lies over a higher order fixed point of the same geometric type as , by proposition 2.5 part (1). Then any -invariant section must vanish at , as well. Varying in a neighborhood of in , these points fill a neighborhood of .
The following result strengthens proposition 2.9 in the special case of holonomy paths coming from –triples. This improvement is crucial, since it provides information on the possible values of invariant sections at the point rather than at .
In the setting of proposition 2.12, suppose that can be chosen so that . Suppose further that is a completely reducible representation of on which acts diagonalizably, and that is the equivariant function corresponding to a -invariant section of . Then .
Proof: Set and , so