Geometric Theory of Weyl Structures
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866917800187002880 |
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| author | Cap, Andreas Mettler, Thomas |
| author_facet | Cap, Andreas Mettler, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1908_10325 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Geometric Theory of Weyl Structures Cap, Andreas Mettler, Thomas Differential Geometry 53A20, 53B15, 53C15, 53C40, 58J60 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. |
| title | Geometric Theory of Weyl Structures |
| topic | Differential Geometry 53A20, 53B15, 53C15, 53C40, 58J60 |
| url | https://arxiv.org/abs/1908.10325 |