Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
On intrinsic geometry of surfaces in normed spaces. / Burago, Dmitri; Ivanov, Sergei.
в: Geometry and Topology, Том 15, № 4, 12.12.2011, стр. 2275-2298.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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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