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- Title
Geometric theory of Weyl structures.
- Authors
Čap, Andreas; Mettler, Thomas
- Abstract
Given a parabolic geometry on a smooth manifold M , we study a natural affine bundle A → 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 T A. 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 | 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.
- Subjects
EINSTEIN, Albert, 1879-1955; STRUCTURAL analysis (Engineering); MONGE-Ampere equations; EINSTEIN manifolds; GEOMETRY; CURVATURE
- Publication
Communications in Contemporary Mathematics, 2023, Vol 25, Issue 7, p1
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199722500262