### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 2275-2298 |

Number of pages | 24 |

Journal | Geometry and Topology |

Volume | 15 |

Issue number | 4 |

DOIs | |

State | Published - Dec 12 2011 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Geometry and Topology*,

*15*(4), 2275-2298. https://doi.org/10.2140/gt.2011.15.2275

}

*Geometry and Topology*, vol. 15, no. 4, pp. 2275-2298. https://doi.org/10.2140/gt.2011.15.2275

**On intrinsic geometry of surfaces in normed spaces.** / Burago, Dmitri; Ivanov, Sergei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On intrinsic geometry of surfaces in normed spaces

AU - Burago, Dmitri

AU - Ivanov, Sergei

PY - 2011/12/12

Y1 - 2011/12/12

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.

UR - http://www.scopus.com/inward/record.url?scp=82955217205&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82955217205&partnerID=8YFLogxK

U2 - 10.2140/gt.2011.15.2275

DO - 10.2140/gt.2011.15.2275

M3 - Article

VL - 15

SP - 2275

EP - 2298

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 4

ER -