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On
intrinsic geometry of surfaces in normed spaces
Dmitri Burago and Sergei Ivanov |
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Geometry & Topology 15 (2011) 2275–2298
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Abstract |
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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 –dimensional space; and (3) geodesics on convex surfaces in a –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
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Mathematical Subject Classification 2010
Primary: 53C22, 53C60
Secondary: 53C45
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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
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Authors |
| Dmitri Burago | |
| Department of Mathematics Pennsylvania State University University Park State College PA 16802 USA |
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| Sergei Ivanov | |
| St Petersburg Department of
Steklov Mathematical Institute Fontanka 27 St Petersburg 191023 Russia |
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| http://eimi.imi.ras.ru/eng/perso/svivanov.php | |
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