Geodesic
On intrinsic geometry of surfaces in normed spaces
Burago, Dmitri1 ; Ivanov, Sergei2
1 Department of Mathematics, Pennsylvania State University, University Park, State College PA 16802, USA
2 St Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St Petersburg, 191023, Russia
Geometry & topology, Tome 15 (2011) no. 4, pp. 2275-2298.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/gt.2011.15.2275
Classification : 53C22, 53C60, 53C45
Keywords: Finsler metric, saddle surface, convex surface, geodesic

Burago, Dmitri 1 ; Ivanov, Sergei 2

1 Department of Mathematics, Pennsylvania State University, University Park, State College PA 16802, USA
2 St Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St Petersburg, 191023, Russia 
@article{GT_2011_15_4_a11,
     author = {Burago, Dmitri and Ivanov, Sergei},
     title = {On intrinsic geometry of surfaces in normed spaces},
     journal = {Geometry & topology},
     pages = {2275--2298},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2011},
     doi = {10.2140/gt.2011.15.2275},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2011.15.2275/}
}
TY  - JOUR
AU  - Burago, Dmitri
AU  - Ivanov, Sergei
TI  - On intrinsic geometry of surfaces in normed spaces
JO  - Geometry & topology
PY  - 2011
SP  - 2275
EP  - 2298
VL  - 15
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.2140/gt.2011.15.2275/
DO  - 10.2140/gt.2011.15.2275
ID  - GT_2011_15_4_a11
ER  - 
%0 Journal Article
%A Burago, Dmitri
%A Ivanov, Sergei
%T On intrinsic geometry of surfaces in normed spaces
%J Geometry & topology
%D 2011
%P 2275-2298
%V 15
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.2140/gt.2011.15.2275/
%R 10.2140/gt.2011.15.2275
%F GT_2011_15_4_a11
Burago, Dmitri; Ivanov, Sergei. On intrinsic geometry of surfaces in normed spaces. Geometry & topology, Tome 15 (2011) no. 4, pp. 2275-2298. doi : 10.2140/gt.2011.15.2275. http://geodesic.mathdoc.fr/articles/10.2140/gt.2011.15.2275/

[1] D Bao, S S Chern, Z Shen, An introduction to Riemann–Finsler geometry, Graduate Texts in Math. 200, Springer (2000)

[2] D Burago, D Grigoriev, A Slissenko, Approximating shortest path for the skew lines problem in time doubly logarithmic in 1/epsilon, Theoret. Comput. Sci. 315 (2004) 371

[3] D Burago, S Ivanov, Isometric embeddings of Finsler manifolds, Algebra i Analiz 5 (1993) 179

[4] Rahul, Mathoverflow.Net/Users/6979, Minimal-length embeddings of braids into $\mathbb{R}^3$ with fixed endpoints, MathOverflow (2010)

[5] S Z Šefel’, The two classes of $k$–dimensional surfaces in $n$–dimensional Euclidean space, Sibirsk. Mat. Ž. 10 (1969) 459

[6] Z Shen, On Finsler geometry of submanifolds, Math. Ann. 311 (1998) 549

Cité par Sources :