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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
On intrinsic geometry of surfaces in normed spaces

Dmitri Burago and Sergei Ivanov

Geometry & Topology 15 (2011) 2275–2298
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.

Keywords
Finsler metric, saddle surface, convex surface, geodesic
Mathematical Subject Classification 2010
Primary: 53C22, 53C60
Secondary: 53C45
References
Publication
Received: 11 January 2011
Revised: 26 May 2011
Accepted: 18 July 2011
Published: 25 November 2011
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Danny Calegari
Authors
Dmitri Burago
Department of Mathematics
Pennsylvania State University
University Park
State College PA 16802
USA
Sergei Ivanov
St Petersburg Department of Steklov Mathematical Institute
Fontanka 27
St Petersburg
191023
Russia
http://eimi.imi.ras.ru/eng/perso/svivanov.php