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Monochromatic Finsler surfaces and a local ellipsoid characterization

Author: Sergei Ivanov
Journal: Proc. Amer. Math. Soc. 146 (2018), 1741-1755
MSC (2010): Primary 52A15, 53B40, 53B25, 52A21
Published electronically: December 18, 2017
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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

Keywords: Finsler surface, Banach-Minkowski space, second fundamental form, ellipsoid characterization
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
Article copyright: © Copyright 2017 American Mathematical Society

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