Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric Theory of Weyl Structures

Andreas Cap, Thomas Mettler

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.

Geometric Theory of Weyl Structures

Abstract

Given a parabolic geometry on a smooth manifold , we study a natural affine bundle , 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 , which induces an almost bi-Lagrangian structure on and a compatible linear connection on . 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 -graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in . 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.

Paper Structure

This paper contains 19 sections, 23 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The bundle of Weyl structures
  3. Parabolic geometries
  4. Weyl structures
  5. The basic geometric structures on A
  6. Relations between natural vector bundles
  7. The Weyl connections
  8. The universal Rho-tensor
  9. Curvature and torsion quantities
  10. The natural almost bi-Lagrangian structure
  11. The almost bi-Lagrangian structure and torsion freeness
  12. Local frames
  13. The canonical metric
  14. Geometry of Weyl structures
  15. Relations to non-linear invariant PDE
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(p:\mathcal{G}\to M,\omega)$ be a parabolic geometry of type $(G,P)$ and $\pi:A\to M$ its associated bundle of Weyl structures. Then the natural 2-form $\Omega\in\Omega^2(A)$ is closed if and only if $(G,P)$ corresponds to a $|1|$-grading and the Cartan geometry $(p:\mathcal{G}\to M,\omega)$ is

Theorems & Definitions (55)

  • Theorem
  • Theorem
  • Corollary
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 45 more