Geometric Theory of Weyl Structures

TL;DR

This paper develops a geometric framework for Weyl structures within parabolic geometries, revealing Einstein metrics and links to Monge-Ampere equations, advancing understanding of geometric structures and their invariants.

Contribution

It introduces a natural affine bundle for Weyl structures, shows the induced Einstein metric in certain cases, and connects Weyl structures to Monge-Ampere equations and convex projective structures.

Findings

The affine bundle of Weyl structures admits a reductive Cartan geometry.

The induced metric is Einstein with non-zero scalar curvature under certain conditions.

Connections to Monge-Ampere equations and convex projective structures are established.

Abstract

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.