Monochromatic Finsler surfaces and a local ellipsoid characterization

Ivanov, Sergei

doi:10.1090/proc/13894

Monochromatic Finsler surfaces and a local ellipsoid characterization
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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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