On intrinsic geometry of surfaces in normed spaces

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.