2017/02/10 by Sergei Ivanov, Ivanov, Sergei
Mathematics · Physics and Astronomy · #52A21 #53B25 #53B40 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1702.03340
openalex publication_date 2017/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the following localized version of a classical ellipsoid characterization: Let B⊂\mathbb R3 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.