arXiv subjects

Andreas Cap

Publications and source records attributed to Andreas Cap.

At least 19 recordsLinked to original sources

arXiv2026

A new construction of the Riemannian deformation sequence

We obtain a new construction of a sequence of invariant differential operators on a Riemannian manifold $(M,g)$ that governs the linearized deformation theory of $g$. Starting from an explicit linear connection on a natural bundle $\mathcal AM\to M$, we construct a twisted de Rham sequence and then apply an analog of the construction of BGG sequences. If $g$ has constant sectional curvature, both sequences are complexes which compute the cohomology of the sheaf of local Killing fields, which are equivalent to parallel sections of $\mathcal AM$. In a second step, we relate the construction to the description of $(M,g)$ as a (torsion-free) Cartan geometry $(\mathcal OM,\omega)$, where $\mathcal OM$ is the orthonormal frame bundle of $M$. This provides a manifest relation of the twisted de Rham sequence to the deformation theory of the Cartan connection $\omega$ (which is easier do deal with than the deformation theory of $g$). The BGG-like construction can then be nicely viewed as interpreting the linearized deformation theory of torsion free Cartan geometries in terms of the underlying Riemannian metric.

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arXiv2026

The Bernstein-Gelfand-Gelfand (BGG) Construction: Algebra, Geometry, and Analysis; Part I

These are lecture notes for the first part of a short course that I taught jointly with Kaibo Hu as part of the thematic program "Differential Complexes: Theory, Discretization, and Applications" at the Erwin Schr\"odinger Institute (ESI) in Vienna. We give an introduction to differential forms and the construction of Berstein-Gelfand-Gelfand (BGG) complexes on open domains in R^n, focusing on the role of representation theory of semisimple Lie groups and Lie algebras in the construction.

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arXiv2026

Projective Infinities and b-Calculus

For a manifold $\overline{M}$ with boundary $\partial M$ and interior $M$, we introduce and study a weakening of the concept of projective compactness for torsion-free linear connections on $M$, which we call projective pre-compactness. Via the Levi-Civita connection, this concept applies to pseudo-Riemannian metrics on $M$. This is motivated by scattering theory and general relativity (GR), via asymptotic forms of metrics used in these areas. In the general setting of a projectively pre-compact connection $\nabla$ we show that, assuming weak asymptotic conditions on the Ricci curvature, there is an induced projective structure on the boundary. Under a slightly stronger condition on Ricci, we show that the standard tractor bundle and its normal tractor connection arise naturally on this boundary structure. The key ingredient to this is that $\nabla$ admits a smooth extension to the boundary as a linear connection on the tensor product of Melrose's b-tangent bundle with a density bundle, which then restricts to the boundary tractor bundle. A projectively pre-compact pseudo-Riemannian metric (satisfying the conditions on the Ricci curvature) is then shown to induce a holonomy reduction of the boundary projective structure to an indefinite orthogonal group. This endows the boundary with a decomposition into so-called curved orbits, which are either open or embedded hypersurfaces, representing space-like, time-like and light-like infinities in a GR context. We introduce and study a new asymptotic form for such metrics which is available near any boundary point and relate it to an asymptotic form used in general relativity, which is only available near boundary points in the open curved orbits. We show that, in that region, projective pre-compactness essentially is equivalent to the asymptotic form from GR, and projective compactness is equivalent to vanishing of the mass aspect.

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arXiv2025

BGG Sequences -- A Riemannian perspective

BGG resolutions and generalized BGG resolutions from representation theory of semisimple Lie algebras have been generalized to sequences of invariant differential operators on manifolds endowed with a geometric structure belonging to the family of parabolic geometries. Two of these structures, conformal structures and projective structures, occur as weakenings of a Riemannian metric respectively of a specified torsion-free connection on the tangent bundle. In particular, one obtains BGG sequences on open subsets of $\mathbb R^n$ as very special cases of the construction. It turned out that several examples of the latter sequences are of interest in applied mathematics, since they can be used to construct numerical methods to study operators relevant for elasticity theory, numerical relativity and related fields. This article is intended to provide an intermediate level between BGG sequences for parabolic geometries and the case of domains in $\mathbb R^n$. We provide a construction of conformal BGG sequences on Riemannian manifolds and of projective BGG sequences on manifolds endowed with a volume preserving linear connection on their tangent bundle. These constructions do not need any input from parabolic geometries. Except from standard differential geometry methods the only deeper input comes from representation theory. So one can either view the results as a simplified version of the constructions for parabolic geometries in an explicit form. Alternatively, one can view them as providing an extension of the simplified constructions for domains in $\Bbb R^n$ to general Riemannian manifolds or to manifolds endowed with an appropriate connection on the tangent bundle.

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arXiv2024

Partial AHS-Structures, their Cartan description and partial BGG sequences

G-structures and Cartan geometries are two major approaches to the description of geometric structures (in the sense of differential geometry) on manifolds of some fixed dimension $n$. We show that both descriptions naturally extend to the setting of manifolds of dimension $\geq n$ which are endowed with a distinguished involutive distribution $F$ of rank $n$. The resulting ``partial'' structures are most naturally interpreted as smooth families of standard G-structures or Cartan geometries on the leaves of the foliation defined by $F$. We prove that for the special class of AHS-structures (also known as $|1|$-graded parabolic geometries) the construction of a canonical Cartan geometry associated to a G-structure extends to this general setting. As an application, we prove that for partial AHS-structures there is an analog of the machinery of BGG sequences. This constructs sequences of differential operators of arbitrarily high order intrinsic to the structures. Under appropriate flatness conditions, these sequence are fine resolutions of sheaves which locally can be realized as pullbacks of sheaves on local leaf spaces for the foliation defined by $F$.

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arXiv2024

Poisson transforms, the BGG complex, and discrete series representations of SU(n+1,1)

The aim of this article is to construct a specific Poisson transform mapping differential forms on the sphere $S^{2n+1}$ endowed with its natural CR structure to forms on complex hyperbolic space. The transforms we construct have values that are harmonic and co-closed and they descend to the BGG (Rumin) complex and intertwine the differential operators in that complex with the exterior derivative. Passing to the Poincar\'e ball model, we analyze the boundary asymptotics of the values of our transforms proving that they admit a continuous extension to the boundary in degrees $\leq n$. Finally, we show that composing the exterior derivative with the transform in degree $n$, one obtains an isomorphism between the kernel of the Rumin operator in degree $n$ and a dense subspace of the $L^2$-harmonic forms on complex hyperbolic space. These are well known to realize the direct sum of all discrete series representations of $SU(n+1,1)$, which we therefore realize on spaces of differential forms on the compact manifold $S^{2n+1}$. The developments in this article are motivated by a program of the third author to prove some instances of the Baum-Connes conjecture. The first part of the article is valid in a much more general setting, and is also relevant for cases in which the conjecture is still open.

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arXiv2023

Induced almost para-K\"ahler Einstein metrics on cotangent bundles

In earlier work we have shown that for certain geometric structures on a smooth manifold $M$ of dimension $n$, one obtains an almost para-K\"ahler--Einstein metric on a manifold $A$ of dimension $2n$ associated to the structure on $M$. The geometry also associates a diffeomorphism between $A$ and $T^*M$ to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-K\"ahler--Einstein metric on $T^*M$. In this short article, we discuss the relation of these metrics to Patterson--Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.

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arXiv2022

Bundles of Weyl structures and invariant calculus for parabolic geometries

For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian and projective geometries. More recently, several general tools were presented for the entire class of parabolic geometries, i.e., the Cartan geometries modelled on homogeneous spaces $G/P$ with $P$ a parabolic subgroup in a semi-simple Lie group $G$. Similarly to conformal Riemannian and projective structures, all these geometries determine a class of distinguished affine connections, which carry an affine structure modelled on differential 1-forms $\Upsilon$. They correspond to reductions of $P$ to its reductive Levi factor, and they are called the Weyl structures similarly to the conformal case. The standard definition of differential invariants in this setting is as affine invariants of these connections, which do not depend on the choice within the class. In this article, we describe a universal calculus which provides an important first step to determine such invariants. We present a natural procedure how to construct all affine invariants of Weyl connections, which depend only tensorially on the deformations $\Upsilon$.

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arXiv2021

A relative mass cocycle and the mass of asymptotically hyperbolic manifolds

We construct a cocycle that, for a given $n$-manifold, maps pairs of asymptotically locally hyperbolic (ALH) metrics to a tractor-valued $(n-1)$-form field on the conformal infinity. This requires the metrics to be asymptotically related to a given order that depends on the dimension. It then provides a local geometric quantity on the boundary that is naturally associated to the pair and can be interpreted as a relative energy-momentum density. It is distinguished as a geometric object by its property of being invariant under suitable diffeomorphisms fixing the boundary, and that act on (either) one of the argument metrics. Specialising to the case of an ALH metric $h$ that is suitably asymptotically related to a locally hyperbolic conformally compact metric, we show that the cocycle determines an absolute invariant $c(h)$, which still is local in nature. This tractor-valued $(n-1)$-form field on the conformal infinity is canonically associated to $h$ (i.e. is not dependent on other choices) and is equivariant under the appropriate diffeomorphisms. Finally specialising further to the case that the boundary is a sphere and that a metric $h$ is asymptotically related to a hyperbolic metric on the interior, we show that the invariant $c(h)$ can be integrated over the boundary. The result pairs with solutions of the KID (Killing initial data) equation to recover the known description of hyperbolic mass integrals of Wang, and Chru\'{s}ciel--Herzlich.

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arXiv2019

Geometric Theory of Weyl Structures

Given a parabolic geometry on a smooth manifold $M$, we study a natural affine bundle $A \to M$, whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on $A$, which induces an almost bi-Lagrangian structure on $A$ and a compatible linear connection on $TA$. We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with non-zero scalar curvature, provided the parabolic geometry is torsion-free and $|1|$-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in $A$. For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampere equation and thus to properly convex projective structures.

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arXiv2019

A Poisson transform adapted to the Rumin complex

Let $G$ be a semisimple Lie group with finite center, $K\subset G$ a maximal compact subgroup, and $P\subset G$ a parabolic subgroup. Following ideas of P.Y.\ Gaillard, one may use $G$-invariant differential forms on $G/K\times G/P$ to construct $G$-equivariant Poisson transforms mapping differential forms on $G/P$ to differential forms on $G/K$. Such invariant forms can be constructed using finite dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on $G/P$ to the associated Bernstein-Gelfand-Gelfand (or BGG) complex in a well defined sense. The main part of the article is devoted to an explicit construction of such transforms with additional favorable properties in the case that $G=SU(n+1,1)$. Thus $G/P$ is $S^{2n+1}$ with its natural CR structure and the relevant BGG complex is the Rumin complex, while $G/K$ is complex hyperbolic space of complex dimension $n+1$. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

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arXiv2018

Parabolic Compactification of Homogeneous Spaces

In this article, we study compactifications of homogeneous spaces coming from equivariant, open embeddings into a generalized flag manifold $G/P$. The key to this approach is that in each case $G/P$ is the homogeneous model for a parabolic geometry; the theory of such geometries provides a large supply of geometric tools and invariant differential operators that can be used for this study. A classical theorem of J.~Wolf shows that any involutive automorphism of a semisimple Lie group $G$ with fixed point group $H$ gives rise to a large family of such compactifications of homogeneous spaces of $H$. Most examples of (classical) Riemannian symmetric spaces as well as many non--symmetric examples arise in this way. A specific feature of the approach is that any compactification of that type comes with the notion of "curved analog" to which the tools we develop also apply. The model example of this is a general Poincar\'e--Einstein manifold forming the curved analog of the conformal compactification of hyperbolic space. In the first part of the article, we derive general tools for the analysis of such compactifications. In the second part, we analyze two families of examples in detail, which in particular contain compactifications of the symmetric spaces $SL(n,\Bbb R)/SO(p,n-p)$ and $SO(n,\Bbb C)/SO(n)$. We describe the decomposition of the compactification into orbits, show how orbit closures can be described as the zero sets of smooth solutions to certain invariant differential operators and prove a local slice theorem around each orbit in these examples.

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arXiv2017

On C-class equations

The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration. While Cartan's definition was in terms of differential invariants being first integrals, all results exhibiting C-classes that we are aware of are based on the fact that a canonical Cartan geometry associated to the equations in the class descends to the space of solutions. For sufficiently low orders, these geometries belong to the class of parabolic geometries and the results follow from the general characterization of geometries descending to a twistor space. In this article we answer the question of whether a canonical Cartan geometry descends to the space of solutions in the case of scalar ODEs of order at least four and of systems of ODEs of order at least three. As in the lower order cases, this is characterized by the vanishing of the generalized Wilczynski invariants, which are defined via the linearization at a solution. The canonical Cartan geometries (which are not parabolic geometries) are a slight variation of those available in the literature based on a recent general construction. All the verifications needed to apply this construction for the classes of ODEs we study are carried out in the article, which thus also provides a complete alternative proof for the existence of canonical Cartan connections associated to higher order (systems of) ODEs.

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arXiv2017

On canonical Cartan connections associated to filtered G-structures

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making it into a nilpotent graded Lie algebra. Assuming that these symbol algebras are the same for all points, one obtains a natural frame bundle for the associated graded to the tangent bundle, and filtered G--structures are defined as reductions of structure group of this bundle. Generalizing the case of parabolic geometries, this article is devoted to the question of whether a filtered G-structure of given type determines a canonical Cartan connection on an extended bundle. As for existence, the result are roughly as general as Morimoto's theorem from 1993, but it has several specific features. First, we allow for general candidates for a homogeneous model and a general version of normalization conditions. Second, the construction is entirely phrased in terms of Lie algebra valued forms and leads to an explicit characterization of the canonical Cartan connection. To verify that the procedure can be applied to a given type of filtered G-structures, only finite dimensional algebraic verifications have to be carried out.

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arXiv2017

Parabolic conformally symplectic structures III; Invariant differential operators and complexes

This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying structure is conformally symplectic (PCS-structures). In that case, we obtained a close relation to parabolic contact structures via a concept of parabolic contactification. It was also shown that special symplectic connections (and thus all connections of exotic symplectic holonomy) arise as the canonical connection of such a structure. In this last part, we use parabolic contactifications and constructions related to Bernstein-Gelfand-Gelfand (BGG) sequences for parabolic contact structures, to construct sequences of differential operators naturally associated to a PCS-structure. In particular, this gives rise to a large family of complexes of differential operators associated to a special symplectic connection. In some cases, large families of complexes for more general instances of PCS-structures are obtained.

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arXiv2016

Parabolic conformally symplectic structures II; parabolic contactification

Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current article, we relate PCS-structures to parabolic contact structures. Starting from a parabolic contact structure with a transversal infinitesimal automorphism, we first construct a natural PCS-structure on any leaf space of the corresponding foliation. Then we develop a parabolic version of contactification to show that any PCS-structure can be locally realized (uniquely up to isomorphism) in this way. In the second part of the paper, these results are extended to an analogous correspondence between contact projective structures and so-called conformally Fedosov structures. The developments in this article provide the technical background for a construction of sequences and complexes of differential operators which are naturally associated to PCS-structures by pushing down BGG sequences on parabolic contact structures. This is the topic of the third part of this series of articles.

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arXiv2016

Parabolic conformally symplectic structures I; definition and distinguished connections

We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading. We show that a structure of each of these types on a smooth manifold $M$ determines a canonical compatible linear connection on the tangent bundle $TM$. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant's theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

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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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