Abstract
In this paper, we study naturally reductive Finsler metrics. We first give a sufficient and necessary condition for a Finsler metric to be naturally reductive with respect to certain transitive group of isometries. Then we study in detail the left invariant naturally reductive metrics on compact Lie groups and give a method to construct the non-Riemannian ones. Further, we give a classification of left invariant naturally reductive metrics on nilpotent Lie groups. Finally, we give a classification of all the naturally reductive Finsler spaces of dimension less or qual to 4. As applications, we obtain some rigidity theorems about naturally reductive Finsler metrics. Namely, any left invariant non-symmetric naturally reductive Finsler metric on a compact simple Lie group or an indecomposable nilpotent Lie group must be Riemannian. On the other hand, we provide a very convenient method to construct non-symmetric Berwald spaces which are neither Riemannian nor locally Minkowskian, a kind of spaces which are sought after in the book by Bao et al. (An introduction to Riemann–Finsler geometry, GTM 200, 2000).
Similar content being viewed by others
References
Akbar-Zadeh H.: Sur les espaces de Finsler á conbures sectionalles constantes. Acad. R. Belg. Bull. CI. Sci. 5(74), 281–322 (1988)
Bao D., Chern S.S., Shen Z.: An Introduction to Riemann–Finsler Geometry. Springer-Verlag, New York (2000)
Berndt J., Vanhecke L.: Geometry of weakly symmetric spaces. J. Math. Soc. Jpn. 48, 745–760 (1996)
Chern S.S., Shen Z.: Riemann–Finsler Geometry. WorldScientific, Singapore (2004)
Deng, S.: An algebraic approach to weakly symmetric Finsler spaces. Can. J. Math. (forthcoming)
Deng S., Hou Z.: Invariant Finsler metrics on homogeneous manifolds. J. Phys. A 37, 8245–8253 (2004)
Deng S., Hou Z.: On locally and globally symmetric Berwald spaces. J. Phys. A 38, 1691–1697 (2005)
D’Atri J.E., Ziller W.: Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Am. Math. Soc. 18(215), 1–72 (1979)
Gordon C.: Naturally reductive homogeneous Riemannian manifolds. Can. J. Math. 37, 467–487 (1985)
Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces, 2nd edn. Academic Press, New York (1978)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol 1, 1963; vol 2, 1969. Interscience Publishers, New York
Kowalski,O.: Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, pp. 131–158. Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale (1983)
Kowalski, O.: Spaces with volume-preserving geodesic symmetries and related classes of Riemannian manifolds, pp. 131–158. Mat. Univ. e Politec. Torino, Fascicolo Speciale (1983)
Kowalski O., Vanhecke L.: Riemanian manifolds with homogeneous geodesics. Boll. UN. Math. Ital. B7(5), 189–246 (1991)
Latifi D.: Homogeneous geodesics on homogeneous Finsler spaces. J. Geom. Phys. 57, 1421–1433 (2007)
Szabó Z.I.: Positive definite Berwald spaces. Tensor (N.S.) 38, 25–39 (1981)
Wilson E.N.: Isometry groups on homogeneous nilmanifolds. Geom. Dedicata 12, 337–346 (1982)
Wolf J.A.: Curvature in nilpotent Lie groups. Proc. AMS 15, 271–274 (1964)
Ziller W.: Weakly symmetric spaces. In: Gindikin, S., Simon, B. (eds) Topics in Geometry: In Memory of Joseph D’Atri, pp. 355–368. Birkhäuser, Boston (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (No.10971104,10621101) of China