TY - JOUR

T1 - On intrinsic geometry of surfaces in normed spaces

AU - Burago, Dmitri

AU - Ivanov, Sergei

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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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U2 - 10.2140/gt.2011.15.2275

DO - 10.2140/gt.2011.15.2275

M3 - Article

AN - SCOPUS:82955217205

VL - 15

SP - 2275

EP - 2298

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 4

ER -