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

In: Geometry and Topology, Vol. 15, No. 4, 12.12.2011, p. 2275-2298.

Research output: Contribution to journal › Article › peer-review

Harvard

Burago, D & Ivanov, S 2011, 'On intrinsic geometry of surfaces in normed spaces', Geometry and Topology, vol. 15, no. 4, pp. 2275-2298. https://doi.org/10.2140/gt.2011.15.2275

APA

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

Vancouver

Burago D, Ivanov S. On intrinsic geometry of surfaces in normed spaces. Geometry and Topology. 2011 Dec 12;15(4):2275-2298. https://doi.org/10.2140/gt.2011.15.2275

Author

Burago, Dmitri ; Ivanov, Sergei. / On intrinsic geometry of surfaces in normed spaces. In: Geometry and Topology. 2011 ; Vol. 15, No. 4. pp. 2275-2298.

BibTeX

@article{d5d8c953da5e43ac9eb0daddebc7d2e6,
title = "On intrinsic geometry of surfaces in normed spaces",
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 = "Convex surface, Finsler metric, Geodesic, Saddle surface",
author = "Dmitri Burago and Sergei Ivanov",
year = "2011",
month = dec,
day = "12",
doi = "10.2140/gt.2011.15.2275",
language = "English",
volume = "15",
pages = "2275--2298",
journal = "Geometry and Topology",
issn = "1465-3060",
publisher = "University of Warwick",
number = "4",

}

RIS

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.

KW - Convex surface

KW - Finsler metric

KW - Geodesic

KW - Saddle surface

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

U2 - 10.2140/gt.2011.15.2275

DO - 10.2140/gt.2011.15.2275

M3 - Article

AN - SCOPUS:82955217205

VL - 15

SP - 2275

EP - 2298

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 4

ER -

ID: 49983630