Monochromatic Finsler surfaces and a local ellipsoid characterization
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- by Sergei Ivanov PDF
- Proc. Amer. Math. Soc. 146 (2018), 1741-1755 Request permission
Abstract:
We prove the following localized version of a classical ellipsoid characterization: Let $B\subset \mathbb {R}^3$ be a convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then all these sections are ellipses and the corresponding part of $B$ is a part of an ellipsoid.
We apply this to differential geometry of Finsler surfaces in normed spaces and show that in certain cases the intrinsic metric of a surface imposes restrictions on its extrinsic geometry similar to implications of Gauss’ Theorema Egregium. As a corollary we construct 2-dimensional Finsler metrics that do not admit local isometric embeddings to dimension 3.
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Additional Information
- Sergei Ivanov
- Affiliation: St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia – and –Saint Petersburg State University, 7/9 Universitetskaya emb., St. Petersburg 199034, Russia
- MR Author ID: 337168
- Email: svivanov@pdmi.ras.ru
- Received by editor(s): February 24, 2017
- Published electronically: December 18, 2017
- Additional Notes: This research was supported by the Russian Science Foundation grant 16-11-10039
- Communicated by: Lei Ni
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1741-1755
- MSC (2010): Primary 52A15, 53B40, 53B25, 52A21
- DOI: https://doi.org/10.1090/proc/13894
- MathSciNet review: 3754357