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Andreas Cap

Publications and source records attributed to Andreas Cap.

At least 37 records · Page 2Linked to original sources
arXiv2016

C-Projective Compactification; (quasi--)Kaehler Metrics and CR boundaries

For complete complex connections on almost complex manifolds we introduce a natural definition of compactification. This is based on almost c--projective geometry, which is the almost complex analogue of projective differential geometry. The boundary at infinity is a (possibly non-integrable) CR structure. The theory applies to almost Hermitean manifolds which admit a complex metric connection of minimal torsion, which means that they are quasi--Kaehler in the sense of Gray--Hervella; in particular it applies to Kaehler and nearly Kaehler manifolds. Via this canonical connection, we obtain a notion of c-projective compactification for quasi--Kaehler metrics of any signature. We describe an asymptotic form for metrics that is necessary and sufficient for c--projective compactness. This metric form provides local examples and, in particular, shows that the usual complete Kaehler metrics associated to smoothly bounded, strictly pseudoconvex domains in C^n are c--projectively compact. For a smooth manifold with boundary and a complete quasi-Kaehler metric $g$ on the interior, we show that if its almost c--projective structure extends smoothly to the boundary then so does its scalar curvature. We prove that $g$ is almost c--projectively compact if and only if this scalar curvature is non-zero on an open dense set of the boundary, in which case it is, along the boundary, locally constant and hence nowhere zero there. Finally we describe the asymptotics of the curvature, showing, in particular, that the canonical connection satisfies an asymptotic Einstein condition. Key to much of the development is a certain real tractor calculus for almost c--projective geometry, and this is developed in the article.

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arXiv2015

Relative BGG sequences; II. BGG machinery and invariant operators

For a real or complex semisimple Lie group $G$ and two nested parabolic subgroups $Q\subset P\subset G$, we study parabolic geometries of type $(G,Q)$. Associated to the group $P$, we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on such bundles. We define a (rather weak) notion of "compressability" for operators acting on relative differential forms with values in a relative tractor bundle. The we develop a general machinery which converts a compressable operator to an operator on bundles associated to completely reducible representations on relative Lie algebra homology groups. Applying this machinery to a specific compressable invariant differential operator of order one, we obtain a relative version of BGG (Bernstein--Gelfand--Gelfand) sequences. All our constructions apply in the case $P=G$, producing new and simpler proofs in the case of standard BGG sequences. We characterize cases in which the relative BGG sequences are complexes or even fine resolutions of certain sheaves and describe these sheaves. We show that this gives constructions of new invariant differential operators as well as of new subcomplexes in certain curved BGG sequences. The results are made explicit in the case of generalized path geometries.

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arXiv2015

Scalar Curvature and Projective Compactness

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth toward infinity. The result implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.

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arXiv2015

Projective Compactness and Conformal Boundaries

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $α$ if the projective structure defined by $\nabla$ smoothly extends to all of $\overline{M}$ in a specific way that depends on no particular choice of boundary defining function. Via the Levi--Civita connection, this concept applies to pseudo--Riemannian metrics on $M$. We study the relation between interior geometry and the possibilities for compactification, and then develop the tools that describe the induced geometry on the boundary. We prove that a pseudo--Riemannian metric on $M$ which is projectively compact of order two admits a certain asymptotic form. This form was known to be sufficient for projective compactness, so the result establishes that it provides an equivalent characterization. From a projectively compact connection on $M$, one obtains a projective structure on $\overline{M}$, which induces a conformal class of (possibly degenerate) bundle metrics on the tangent bundle to the hypersurface $\partial M$. Using the asymptotic form, we prove that in the case of metrics, which are projectively compact of order two, this boundary structure is always non--degenerate. We also prove that in this case the metric is necessarily asymptotically Einstein, in a natural sense. Finally, a non--degenerate boundary geometry gives rise to a (conformal) standard tractor bundle endowed with a canonical linear connection, and we explicitly describe these in terms of the projective data of the interior geometry.

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arXiv2014

Projective Compactifications and Einstein metrics

For complete affine manifolds we introduce a definition of compactification based on the projective differential geometry (i.e.\ geodesic path data) of the given connection. The definition of projective compactness involves a real parameter $α$ called the order of projective compactness. For volume preserving connections, this order is captured by a notion of volume asymptotics that we define. These ideas apply to complete pseudo-Riemannian spaces, via the Levi-Civita connection, and thus provide a notion of compactification alternative to conformal compactification. For each order $α$, we provide an asymptotic form of a metric which is sufficient for projective compactness of the given order, thus also providing many local examples. Distinguished classes of projectively compactified geometries of orders one and two are associated with Ricci-flat connections and non--Ricci--flat Einstein metrics, respectively. Conversely, these geometric conditions are shown to force the indicated order of projective compactness. These special compactifications are shown to correspond to normal solutions of classes of natural linear PDE (so-called first BGG equations), or equivalently holonomy reductions of projective Cartan/tractor connections. This enables the application of tools already available to reveal considerable information about the geometry of the boundary at infinity. Finally, we show that metrics admitting such special compactifications always have an asymptotic form as mentioned above.

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arXiv2014

Pushing down the Rumin complex to conformally symplectic quotients

Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism $ξ$, we show that any local leaf space $M$ for the foliation determined by $ξ$ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on $M$, whose cohomology can be computed. Applying this construction locally, one obtains a complex intrinsically associated to any manifold endowed with a cs-structure, which recovers the generalization of the so-called Rumin-Seshadri complex to the conformally symplectic setting. The cohomology of this more general complex can be computed using the push-down construction.

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arXiv2014

Holonomy reductions of Cartan geometries and curved orbit decompositions

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a `curved orbit decomposition'. The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal and CR-geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third order differential equation that governs the existence of a compatible Einstein metric. In CR-geometry we discuss an invariant equation that governs the existence of a compatible Kähler-Einstein metric.

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arXiv2009

Equivariant quantizations for AHS--structures

We construct an explicit scheme to associate to any potential symbol an operator acting between sections of natural bundles (associated to irreducible representations) for a so-called AHS-structure. Outside of a finite set of critical (or resonant) weights, this procedure gives rise to a quantization, which is intrinsic to this geometric structure. In particular, this provides projectively and conformally equivariant quantizations for arbitrary symbols on general (curved) projective and conformal structures.

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arXiv2009

Conformally Invariant Operators via Curved Casimirs: Examples

We discuss a general scheme for a construction of linear conformally invariant differential operators from curved Casimir operators; we then explicitly carry this out for several examples. Apart from demonstrating the efficacy of the approach via curved Casimirs, this shows that this method applies both in regular and in singular infinitesimal character, and also that it can be used to construct standard as well as non--standard operators. The examples treated include conformally invariant operators with leading term, in one case, a square of the Laplacian, and in another case, a cube of the Laplacian.

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arXiv2008

CR-Tractors and the Fefferman Space

We develop the natural tractor calculi associated to conformal and CR structures as a fundamental tool for the study of Fefferman's construction of a canonical conformal class on the total space of a circle bundle over a non--degenerate CR manifold of hypersurface type. In particular we construct and treat the basic objects that relate the natural bundles and natural operators on the two spaces. This is illustrated with several applications: We prove that a number of conformally invariant overdetermined systems admit non--trivial solutions on any Fefferman space. We show that the space of conformal Killing fields on a Fefferman space naturally decomposes into a direct sum of subspaces, which admit an interpretaion as solutions of certain CR invariant PDE's. Finally we explicitly analyze the relation between tractor calculus on a CR manifold and the complexified conformal tractor calculus on its Fefferman space, thus obtaining a powerful computational tool for working with the Fefferman construction.

math.DG↗
arXiv2008

Contact projective structures and chains

Contact projective structures have been profoundly studied by D.J.F. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox' construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman's construction of a conformal structure associated to a CR structure. We show that, on the level of Cartan connections, this Fefferman--type construction is compatible with normality if and only if the initial structure has vanishing contact torsion. This leads to a geometric description of the paths that have to be added to the contact geodesics of a contact projective structure in order to obtain the subordinate projective structure. They are exactly the chains associated to the contact projective structure, which are analogs of the Chern-Moser chains in CR geometry. Finally, we analyze the consequences for the geometry of chains and prove that a chain-preserving contactomorphism must be a morphism of contact projective structures.

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arXiv2008

AHS-structures and affine holonomies

We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS-structures on certain generalized flag manifolds.

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arXiv2008

On automorphism groups of some types of generic distributions

To certain types of generic distributions (subbundles in a tangent bundle) one can associate canonical Cartan connections. Many of these constructions fall into the class of parabolic geometries. The aim of this article is to show how strong restrictions on the possibles sizes of automorphism groups of such distributions can be deduced from the existence of canonical Cartan connections. This needs no information on how the Cartan connections are actually constructed and only very basic information on their properties. In particular, we discuss the examples of generic distributions of rank two in dimension five, rank three in dimension six, and rank four in dimension seven.

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arXiv2007

Curved Casimir Operators and the BGG Machinery

We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence.

math.DG↗
arXiv2007

On Nurowski's conformal structure associated to a generic rank two distribution in dimension five

For a generic distribution of rank two on a manifold $M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly constructed a pseudo--Riemannian metric on $M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution. In the second part of the article, we relate this conformal class to the canonical Cartan connection associated to the distribution. This is used to prove that it coincides with the conformal class constructed by Nurowski.

math.DG↗
arXiv2006

A holonomy characterisation of Fefferman spaces

We prove that Fefferman spaces, associated to non--degenerate CR structures of hypersurface type, are characterised, up to local conformal isometry, by the existence of a parallel orthogonal complex structure on the standard tractor bundle. This condition can be equivalently expressed in terms of conformal holonomy. Extracting from this picture the essential consequences at the level of tensor bundles yields an improved, conformally invariant analogue of Sparling's characterisation of Fefferman spaces.

math.DG↗
arXiv2006

Overdetermined systems, conformal geometry, and the BGG complex

This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work with T.P. Branson, M.G. Eastwood, and A.R. Gover. First a simple special case is discussed, then the (Riemannian) procedure is described in general. The prolongation procedure was derived from a simplification of the construction of Bernstein-Gelfand-Gelfand (BGG) sequences of invariant differential operators for certain geometric structures. The version of this construction for conformal structures is described next. Finally, we discuss generalizations of both the prolongation procedure and the construction of invariant operators to other geometric structures.

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