Monochromatic Finsler surfaces and a local ellipsoid characterization

TL;DR

This paper establishes a local ellipsoid characterization for convex bodies with smooth boundaries and applies it to Finsler surface geometry, revealing restrictions on embeddings and intrinsic-extrinsic metric relations.

Contribution

It introduces a localized ellipsoid characterization theorem and applies it to Finsler surfaces, showing intrinsic metrics can restrict extrinsic embeddings.

Findings

Sections of convex bodies are ellipses under certain conditions

Intrinsic metrics can restrict extrinsic geometry of Finsler surfaces

Constructs Finsler metrics without local isometric embeddings in 3D

Abstract

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be 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.