Geometric Theory of Weyl Structures

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.

39 pages, final version, to appear in Commun. Contemp. Math

References in corpus (3)

• Ricci-corrected derivatives and invariant differential operators
• Convex projective surfaces with compatible Weyl connection are hyperbolic
• Holomorphic differentials, thermostats and Anosov flows

Cited by in corpus (2)

• Induced almost para-Kähler Einstein metrics on cotangent bundles
• Bundles of Weyl structures and invariant calculus for parabolic geometries