### 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 - 2011 |

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

## Fingerprint Dive into the research topics of 'On intrinsic geometry of surfaces in normed spaces'. Together they form a unique fingerprint.

## Cite this

Burago, D., & Ivanov, S. (2011). On intrinsic geometry of surfaces in normed spaces.

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