We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic orbit Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.
Nous montrons des résultats de rigidité sur les espaces de Finsler homogènes, dont les géodésiques sont des espaces homogènes à courbure non positive. En particulier, nous montrons qu'un tel espace de Finsler dont la courbure de drapeau est strictement négative doit être un espace symétrique riemannien non compact de rang un.
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@article{CRMATH_2017__355_9_987_0,
author = {Xu, Ming and Deng, Shaoqiang},
title = {Rigidity of negatively curved geodesic orbit {Finsler} spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {987--990},
publisher = {Elsevier},
volume = {355},
number = {9},
year = {2017},
doi = {10.1016/j.crma.2017.09.003},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.crma.2017.09.003/}
}
TY - JOUR AU - Xu, Ming AU - Deng, Shaoqiang TI - Rigidity of negatively curved geodesic orbit Finsler spaces JO - Comptes Rendus. Mathématique PY - 2017 SP - 987 EP - 990 VL - 355 IS - 9 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2017.09.003/ DO - 10.1016/j.crma.2017.09.003 LA - en ID - CRMATH_2017__355_9_987_0 ER -
%0 Journal Article %A Xu, Ming %A Deng, Shaoqiang %T Rigidity of negatively curved geodesic orbit Finsler spaces %J Comptes Rendus. Mathématique %D 2017 %P 987-990 %V 355 %N 9 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2017.09.003/ %R 10.1016/j.crma.2017.09.003 %G en %F CRMATH_2017__355_9_987_0
Xu, Ming; Deng, Shaoqiang. Rigidity of negatively curved geodesic orbit Finsler spaces. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 987-990. doi : 10.1016/j.crma.2017.09.003. https://www.numdam.org/articles/10.1016/j.crma.2017.09.003/
[1] Riemannian flag manifolds with homogeneous geodesics, Trans. Amer. Math. Soc., Volume 359 (2007), pp. 3769-3789
[2] Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3), Volume 15 (1961), pp. 179-246
[3] Homogeneous Finsler Spaces, Springer, New York, 2012
[4] Invariant Finsler metrics on homogeneous manifolds, J. Phys. A, Math. Gen., Volume 37 (2004), pp. 8245-8253
[5] Homogeneous Finsler spaces of negative curvature, J. Geom. Phys., Volume 57 (2007), pp. 657-664
[6] Clifford–Wolf translations of Finsler spaces, Forum Math., Volume 26 (2014), pp. 1413-1428
[7] Homogeneous Riemannian manifolds whose geodesics are orbits, Topics in Geometry, in Memory of Joseph D'Atri, Birkhäuser, 1996, pp. 155-174
[8] On homogeneous manifolds of negative curvature, Math. Ann., Volume 211 (1974), pp. 23-34
[9] On the fundamental equations of homogeneous Finsler spaces, Differ. Geom. Appl., Volume 40 (2015), pp. 187-208
[10] On some types of topological groups, Ann. of Math., Volume 50 (1949), pp. 507-558
[11] Lectures on Finsler Geometry, World Scientific Publishing, 2001
[12] H. Wang, J. Li, Cartan's fixed points theorem for Finsler spaces, preprint.
[13] Homogeneity and bounded isometries in manifolds of negative curvature, Ill. J. Math., Volume 8 (1964), pp. 14-18
[14] Normal homogeneous Finsler spaces, Transform. Groups (2017) (published online) | DOI
[15] Even dimensional homogeneous Finsler spaces with positive flag curvature, Indiana Univ. Math. J., Volume 66 (2017), pp. 949-972
[16] Finsler spaces whose geodesics are orbits, Differ. Geom. Appl., Volume 36 (2014), pp. 1-23
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