#### Volume 15, issue 4 (2011)

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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
##### 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