On intrinsic geometry of surfaces in normed spaces

TL;DR

This paper explores the intrinsic geometry of surfaces in normed spaces, revealing properties of geodesics on saddle and convex surfaces, and showing local embeddings of Finsler manifolds as saddle surfaces.

Contribution

It establishes new results on geodesic behavior and embeddings of surfaces in Minkowski and Finsler spaces, enhancing understanding of their intrinsic geometry.

Findings

Geodesics on saddle surfaces have no conjugate points and minimize length.

Any 2D Finsler manifold can be locally embedded as a saddle surface in 4D.

Geodesics on convex surfaces in 3D do not globally minimize length.

Abstract

We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension) behave as they are expected to: they have no conjugate points and thus minimize length in their homotopy class; (2) in contrast, every two-dimensional Finsler manifold can be locally embedded as a saddle surface in a 4-dimensional space; and (3) geodesics on convex surfaces in a 3-dimensional space also behave as they are expected to: on a complete strictly convex surface, no complete geodesic minimizes the length globally.