On intrinsic geometry of surfaces in normed spaces

Details On intrinsic geometry of surfaces in normed spaces On intrinsic geometry of surfaces in normed spaces

2010-11-25

arxiv.org/abs/1011.5670v2

Geometry & Topology 15 (2011), no. 4, 2275--2298

Mathematics

Differential Geometry

18 pages, v2: corrected a few misprints

Scientific paper

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.

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