Geometric theory of Weyl structures

Abstract

Given a parabolic geometry on a smooth manifold [Formula: see text], we study a natural affine bundle [Formula: see text], 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 [Formula: see text], which induces an almost bi-Lagrangian structure on [Formula: see text] and a compatible linear connection on [Formula: see text]. We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with nonzero scalar curvature, provided that the parabolic geometry is torsion-free and [Formula: see text]-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in [Formula: see text]. 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.