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On symmetric Finsler spaces. (English) Zbl 1141.53071

Summary: In this paper, we study symmetric Finsler spaces. We first study some geometric properties of globally symmetric Finsler spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Then, we prove that a globally symmetric Finsler space is a Berwald space. As an application, we use the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Finsler spaces and obtain the formulas for flag curvature and Ricci scalar. Finally, some rigidity results of locally symmetric Finsler spaces related to the flag curvature are also given.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C35 Differential geometry of symmetric spaces
53C24 Rigidity results
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[1] W. Ambrose and I. M. Singer, A theorem on holonomy, Transactions of the American Mathematical Society 75 (1953), 428–443. · Zbl 0052.18002
[2] P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler spaces with applications in Physics and Biology, Kluwer Academic Publishers, Dordrecht, 1993. · Zbl 0821.53001
[3] D. Bao and S. S. Chern, On a notable connection in Finsler geometry, Houston Journal of Mathematics 19 (1993), 135–180. · Zbl 0787.53018
[4] D. Bao, S. S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, New York, 2000. · Zbl 0954.53001
[5] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, Journal of Differential Geometry 66 (2004), 377–435. · Zbl 1078.53073
[6] H. Busemann and B. Phadke, Two theorems on general symmetric spaces, Pacific Journal of Mathematics 92 (1981), 39–48. · Zbl 0458.53031
[7] S. S. Chern, Z. Shen,, Riemann-Finsler Geometry, WorldScientific, Singapore, 2004.
[8] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific Journal of Mathematics 207 (2002), 149–155. · Zbl 1055.53055
[9] S. Deng and Z. Hou, Invariant Finsler metrics on homogeneous manifolds, Journal of Physics A: Mathematical and General 37 (2004), 8245–8253. · Zbl 1062.58007
[10] S. Deng and Z. Hou, On locally and globally symmetric Berwald spaces, Journal of Physics A: Mathematical and General 38 (2005), 1691–1697.
[11] S. Deng and Z. Hou, Minkowski symmetric Lie algebras and symmetric Berwald spaces, Geometriae Dedicata 113 (2005), 95–105. · Zbl 1084.53059
[12] P. Foulon, Curvature and global rigidity in Finsler manifolds, Houston Journal of Mathematics 28 (2002), 263–292. · Zbl 1027.53088
[13] S. Helgason, Differential Geometry, Lie groups and Symmetric Spaces, 2nd ed., Academic Press, 1978. · Zbl 0451.53038
[14] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, Vol. 1, 1963, Vol. 2, 1969. · Zbl 0119.37502
[15] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 1009.53004
[16] Z. I. Szab√≥, Positive Definite Berwald Spaces, The Tensor Society, Tensor New Series 38 (1981), 25–39.
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