arXiv subjects

Katharina Neusser

Publications and source records attributed to Katharina Neusser.

13 recordsLinked to original sources

arXiv2026

Geometry of conic connections

A cone structure on a complex manifold $M$ is a closed submanifold $\mathcal C\subset \mathbb P TM$ of the projectivized tangent bundle of $M$ that is submersive over $M$. So this defines a set $\mathcal C_x$ of distinguished directions in each point $x\in M$. A conic connection on $\mathcal C$ is then a family of unparametrized curves on $M$ that comprises exactly one curve through each point $x\in M$ in each direction in $\mathcal C_x\subset \mathbb PT_xM$. This can be encoded by a line subbundle $\mathcal F\subset T\mathcal C$. The subclass of characteristic conic connections is defined by the vanishing of a simple invariant, called characteristic torsion. For those, one has a much more subtle and slightly mysterious invariant called the cubic torsion. The first aim of this article is to provide a new approach to the cubic torsion, which also leads to a geometric condition characterizing its vanishing. We then specialize to the case of isotrivial cone structures, for which the fibers of $\mathcal C$ are assumed to be of some fixed type. Such a structure induces a first-order $G$-structure on $M$ whose structure group is the projective automorphism group of the model fiber. Moreover, any connection $γ$ on the associated $G$-structure induces a conic connection $\mathcal F^γ$ on $\mathcal C$. Assuming that the model fiber is homogeneous, we study the relation between the torsion and curvature of a connection $γ$ and the characteristic and cubic torsion of $\mathcal F^γ$. As an application we discuss cone structures of subadjoint type, showing in particular that there are such structures admitting conic connections with vanishing characteristic and cubic torsion that are not locally flat.

math.DG↗

arXiv2026

An Embedding Theorem for Tractor Bundles, and an Application in Conformal Pseudo-Riemannian Geometry

We provide an extension of the Gromov-Zimmer embedding theorem for Cartan geometries of [Bader U., Frances C., Melnick K., Geom. Funct. Anal. 19 (2009), 333-355, arXiv:0709.3844] to tractor bundles carrying any invariant connection, including tractor connections and prolongation connections of first BGG operators for parabolic geometries. As an application, we prove a rigidity result for conformal actions of special pseudo-unitary groups on closed, simply connected, analytic pseudo-Riemannian manifolds.

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arXiv2022

Projective geometry of 3-Sasaki structures

We show that $3$-Sasaki structures admit a natural description in terms of projective differential geometry. This description provides a concrete link between $3$-Sasaki structures and several other geometries and constructions via a single unifying picture. First we establish that a $3$-Sasaki structure may be understood as a projective structure equipped with a certain holonomy reduction to the (possibly indefinite) unitary quaternionic group $\textrm{Sp}(p,q)$, namely a parallel hyperkähler structure on the projective tractor bundle satisfying a particular genericity condition. For the converse, where one begins with a general parallel hyperkähler structure on the projective tractor bundle, the genericity condition is not automatic. Indeed we prove that generically such a reduction decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) $3$-Sasaki structures and a closed separating hypersurface at infinity with respect to the $3$-Sasaki metrics. Moreover, it is shown that the latter hypersurface inherits a Biquard-Fefferman conformal structure, which thus (locally) fibres over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the $3$-Sasaki structures on the open manifolds. As an application we describe the projective compactification of (suitably) complete, non-compact (indefinite) $3$-Sasaki manifolds and recover Biquard's notion of asymptotically hyperbolic quaternionic Kähler metrics.

math.DG↗

arXiv2020

Cone structures and parabolic geometries

A cone structure on a complex manifold $M$ is a closed submanifold $\mathcal C \subset \mathbb P TM$ of the projectivized tangent bundle which is submersive over $M$. A conic connection on $\mathcal C$ specifies a distinguished family of curves on $M$ in the directions specified by $\mathcal C$. There are two common sources of cone structures and conic connections, one in differential geometry and another in algebraic geometry. In differential geometry, we have cone structures induced by the geometric structures underlying holomorphic parabolic geometries, a classical example of which is the null cone bundle of a holomorphic conformal structure. In algebraic geometry, we have the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. The local invariants of the cone structures in parabolic geometries are given by the curvature of the parabolic geometries, the nature of which depend on the type of the parabolic geometry, i.e., the type of the fibers of $\mathcal C \to M$. For the VMRT-structures, more intrinsic invariants of the conic connections which do not depend on the type of the fiber play important roles. We study the relation between these two different aspects for the cone structures induced by parabolic geometries associated with a long simple root of a complex simple Lie algebra. As an application, we obtain a local differential-geometric version of the global algebraic-geometric recognition theorem due to Mok and Hong--Hwang. In our local version, the role of rational curves is completely replaced by appropriate torsion conditions on the conic connection.

math.DG↗

arXiv2019

Projective geometry of Sasaki-Einstein structures and their compactification

We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete non-compact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the Kähler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail.

math.DG↗

arXiv2018

On the groups of c-projective transformations of complete Kähler manifolds

We show that for any complete connected Kähler manifold the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini-Study metric. This establishes a stronger version of the recently proved Yano-Obata conjecture for complete Kähler manifolds.

math.DG↗

arXiv2016

C-projective geometry

We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kaehler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kaehler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano-Obata conjecture for complete Kaehler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini-Study metric.

math.DG↗

arXiv2016

A canonical connection on sub-Riemannian contact manifolds

We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.

math.DG↗

arXiv2015

Strongly essential flows on irreducible parabolic geometries

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence of a strongly essential flow implies local flatness of the geometry on an open set having the fixed point in its closure. For almost c-projective and almost quaternionic structures we can moreover show flatness of the geometry on a neighborhood of the fixed point.

math.DG↗

arXiv2012

Some differential complexes within and beyond parabolic geometry

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to certain geometries beyond the parabolic realm.

math.DG↗

arXiv2012

Prolongation on regular infinitesimal flag manifolds

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-)linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds $M$ a conceptual method to rewrite these systems as systems of the form $\tilde\nabla(Σ)+C(Σ)=0$, where $\tilde\nabla$ is a linear connection on some vector bundle $V$ over $M$ and $C: V\rightarrow T^*M\otimes V$ is a (vector) bundle map. In particular, if the overdetermined system is linear, $\tilde\nabla+C$ will be a linear connection on $V$ and hence the dimension of its solution space is bounded by the rank of $V$. We will see that the rank of $V$ can be easily computed using representation theory.

math.DG↗

arXiv2009

Universal Prolongation of Linear Partial Differential Equations on Filtered Manifolds

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that the solution space of such equations is always finite dimensional.

math.DG↗

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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