arXiv subjects

Katja Sagerschnig

Publications and source records attributed to Katja Sagerschnig.

15 recordsLinked to original sources

arXiv2024

Conformal structures with $G_2$-symmetric twistor distribution

For any 4D split-signature conformal structure, there is an induced twistor distribution on the 5D space of all self-dual totally null 2-planes, which is $(2,3,5)$ when the conformal structure is not anti-self-dual. Several examples where the twistor distribution achieves maximal symmetry (the split-real form of the exceptional simple Lie algebra of type $\mathrm{G}_2$) were previously known, and these include fascinating examples arising from the rolling of surfaces without twisting or slipping. Relaxing the rolling assumption, we establish a complete local classification result among those homogeneous 4D split-conformal structures for which the symmetry algebra induces a multiply-transitive action on the 5D space. Furthermore, we discuss geometric properties of these conformal structures such as their curvature, holonomy, and existence of Einstein representatives.

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arXiv2023

Parabolic quasi-contact cone structures with an infinitesimal symmetry

We interpret the property of having an infinitesimal symmetry as a variational property in certain geometric structures. This is achieved by establishing a one-to-one correspondence between a class of cone structures with an infinitesimal symmetry and geometric structures arising from certain systems of ODEs that are variational. Such cone structures include pseudo-Riemannian conformal structures and distributions of growth vector (2,3,5) and (3,6). The correspondence is obtained via symmetry reduction and quasi-contactification. Subsequently, for each class of such cone structures we provide invariant conditions that imply more specific properties, such as having a null infinitesimal symmetry, being foliated by null submanifolds, or having reduced holonomy to the appropriate contact parabolic subgroup. As an application of our results we give an alternative proof of the variationality of chains in CR geometry.

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arXiv2023

Modified conformal extensions

We present a geometric construction and characterization of $2n$-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson--Walker metric construction for $n$-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman--Graham ambient metric and show vanishing of $Q$-curvature.

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arXiv2020

Conformal Patterson-Walker metrics

The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson-Walker metric. Finally, we describe all symmetries of the conformal Patterson-Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure.

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arXiv2018

The geometry of marked contact Engel structures

A contact twisted cubic structure (M,C,S) is a 5-dimensional manifold M together with a contact distribution C and a bundle S of twisted cubics that is compatible with the conformal symplectic form on C. In Engel's classical work, the Lie algebra of the exceptional Lie group G_2 was realized as the symmetry algebra of the most symmetric contact twisted cubic structure; we thus refer to this one as the contact Engel structure. In the present paper we equip the contact Engel structure with a smooth section s: M-> S that `marks' a point in each twisted cubic. We study the local geometry of the resulting structures (M,C,S,s), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of M by curves whose tangent directions are everywhere contained in S. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension greater than 5, and we prove an analogue of the classical Kerr theorem from relativity.

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arXiv2017

The almost Einstein operator for $(2, 3, 5)$ distributions

For the geometry of oriented $(2, 3, 5)$ distributions $(M, {\mathbf D})$, which correspond to regular, normal parabolic geometries of type $(\mathrm{G}_2, P)$ for a particular parabolic subgroup $P < \mathrm{G}_2$, we develop the corresponding tractor calculus and use it to analyze the first BGG operator $Θ_0$ associated to the $7$-dimensional irreducible representation of $\mathrm{G}_2$. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator are automatically normal, yielding a geometric interpretation of $\ker Θ_0$: For any $(M, {\mathbf D})$, this kernel consists precisely of the almost Einstein scales of the Nurowski conformal structure on $M$ that ${\mathbf D}$ determines. We apply our formula for $Θ_0$ (1) to recover efficiently some known solutions, (2) to construct a distribution with root type $[3, 1]$ with a nonzero solution, and (3) to show efficiently that the conformal holonomy of a particular $(2, 3, 5)$ conformal structure is equal to $\mathrm{G}_2$.

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arXiv2017

A Projective-to-Conformal Fefferman-Type Construction

We study a Fefferman-type construction based on the inclusion of Lie groups ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.

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arXiv2017

Contact manifolds, Lagrangian Grassmannians and PDEs

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n+1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a 30-hours Ph.D course given by two of the authors (GM and GM). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

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arXiv2017

The Geometry of Almost Einstein (2,3,5) Distributions

We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures $\mathbf{c}$ that are induced by at least two distinct oriented (2,3,5) distributions; in this case there is a 1-parameter family of such distributions that induce $\mathbf{c}$. Second, they are characterized by the existence of a holonomy reduction to $SU(1,2)$, $SL(3,{\mathbb R})$, or a particular semidirect product $SL(2,{\mathbb R})\ltimes Q_+$, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2,3,5) distributions and many other geometries - several classical geometries among them - including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; Kähler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional Kähler-Einstein or para-Kähler-Einstein structure a family of (2,3,5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2,3,5) conformal structures for which the Einstein constant is positive and negative.

math.DG↗

arXiv2016

Fefferman-Graham ambient metrics of Patterson-Walker metrics

Given an $n$-dimensional manifold $N$ with an affine connection $D$, we show that the associated Patterson-Walker metric $g$ on $T^*N$ admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which the ambient metric exists to all orders and can be realized in a natural and explicit way. In particular, it follows that Patterson-Walker metrics have vanishing Fefferman-Graham obstruction tensors. As an application of the concrete ambient metric realization we show in addition that Patterson-Walker metrics have vanishing Q-curvature.

math.DG↗

arXiv2016

New relations between $G_2$-geometries in dimensions 5 and 7

There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a seven-manifold. We present a natural geometric construction of a Lie contact structure on a seven-dimensional bundle over a five-manifold endowed with a $(2,3,5)$-distribution. For a class of distributions the induced Lie contact structure is constructed explicitly and we determine its symmetries. We further study the relation between the canonical normal Cartan connections associated with the two structures. In particular, we show that the Cartan holonomy of the induced Lie contact structure reduces to $G_2$. Moreover, the curved orbit decomposition associated with a $\mathrm{G}_2$-reduced Lie contact structure on a seven-manifold is discussed. It is shown that in a neighbourhood of each point on the open curved orbit the structure descends to a $(2,3,5)$-distribution on a local leaf space, provided an additional curvature condition is satisfied. The closed orbit carries an induced $G_2$-contact structure.

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arXiv2011

A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors

We treat a non-normal Fefferman-type construction based on an inclusion $\SL(n+1)\embed\Spin(n+1,n+1)$. The construction associates a split signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. For $n\geq 3$ the induced conformal Cartan connection is shown to be normal if and only if it is flat. The main technical work of this article consists in showing that in the non-flat case the normalised conformal Cartan connection still allows a parallel (pure) spin-tractor and thus a corresponding (pure) twistor spinor on the conformal space. The Fefferman-type construction presented here is an alternative approach to study a construction of Dunajski-Tod

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arXiv2010

The twistor spinors of generic 2- and 3-distributions

Generic distributions on 5- and 6-manifolds give rise to conformal structures that were discovered by P. Nurowski resp. R. Bryant. We describe both as Fefferman-type constructions and show that for orientable distributions one obtains conformal spin structures. The resulting conformal spin geometries are then characterized by their conformal holonomy and equivalently by the existence of a twistor spinor which satisfies a genericity condition. Moreover, we show that given such a twistor spinor we can decompose a conformal Killing field of the structure. We obtain explicit formulas relating conformal Killing fields, almost Einstein structures and twistor spinors.

math.DG↗

arXiv2009

Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decomposition

Given a maximally non-integrable 2-distribution ${\mathcal D}$ on a 5-manifold $M$, it was discovered by P. Nurowski that one can naturally associate a conformal structure $[g]_{\mathcal D}$ of signature (2,3) on $M$. We show that those conformal structures $[g]_{\mathcal D}$ which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of $[g]_{\mathcal D}$ can be decomposed into a symmetry of ${\mathcal D}$ and an almost Einstein scale of $[g]_{\mathcal D}$.

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.

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