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An Algebraic Approach to Weakly Symmetric Finsler Spaces

Part of: Lie groups Global differential geometry Infinite-dimensional manifolds

Published online by Cambridge University Press:  20 November 2018

Shaoqiang Deng
Shaoqiang Deng*
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China, e-mail: dengsq@nankai.edu.cn
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Abstract

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In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing $\text{S}$-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing $\text{S}$-curvature may exist at large. Hence the generalized volume comparison theorems due to $\text{Z}$. Shen are valid for a rather large class of Finsler spaces.

Keywords

weakly symmetric Finsler spacesweakly symmetric Lie algebrasBerwald spacesS-curvature

MSC classification

Secondary: 53C60: Finsler spaces and generalizations (areal metrics) 58B20: Riemannian, Finsler and other geometric structures 22E46: Semisimple Lie groups and their representations 22E60: Lie algebras of Lie groups
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 1 , 01 February 2010 , pp. 52 - 73
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] D., Bao and S. S., Chern, On a notable connection in Finsler geometry. Houston J. Math. 19(1993), no. 1, 135-180.Google Scholar
[2] D., Bao, S. S., Chern and Z., Shen, An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics 200. Springer-Verlag, New York, 2000.Google Scholar
[3] J., Berndt, O., Kowalski, and L., Vanhecke, Geodesics in weakly symmetric spaces. Ann. Global Anal. Geom. 15(1997), no. 2, 153-156. doi:10.1023/A:1006565909527Google Scholar
[4] J., Berndt and L., Vanhecke, Geometry of weakly symmetric spaces. J. Math. Soc. Japan 48(1996), no. 4, 745-760. doi:10.2969/jmsj/04840745Google Scholar
[5] S. S., Chern and Z., Shen, Riemann-Finsler Geometry. Nankai Tracts in Mathematics 6. World Scientific Publishers, Hackensak, NJ, 2005.Google Scholar
[6] S., Deng and Z., Hou, The group of isometries of a Finsler space. Pacific J. Math. 207(2002), no. 1, 149-155. doi:10.2140/pjm.2002.207.149Google Scholar
[7] S., Deng and Z., Hou, Invariant Finsler metrics on homogeneousmanifolds. J. Phys. A 37(2004), no. 34, 8245-8253. doi:10.1088/0305-4470/37/34/004Google Scholar
[8] S., Deng and Z., Hou, On symmetric Finsler spaces. Israel J. Math. 162(2007), 197-219. doi:10.1007/s11856-007-0095-6Google Scholar
[9] S., Deng and Z., Hou, Weakly symmetric Finsler spaces. To appear in Commun. Contemp. Math. doi:10.1007/s11856-007-0095-6Google Scholar
[10] S., Helgason, Differential Geometry, Lie groups and Symmetric Spaces. Second edition. Pure and Applied Mathematics 80. Academic Press, New York, 1978.Google Scholar
[11] S., Kobayashi, Homogeneous Riemannian manifolds of negative curvature. Tôhoku Math. J. 14(1962), 413-415. doi:10.2748/tmj/1178244077Google Scholar
[12] S., Kobayashi, and K., Nomizu, Foundations of Differential Geometry Vol. 1 and 2. Interscience Publishers, New York, 1963, 1969.Google Scholar
[13] O., Kowalski, Spaces with volume-preserving symmetries and related classes of Riemannian manifolds Rend. Sem.Mat. Univ. Politec. Torino, Facicolo Speciale (1983), 131-158.Google Scholar
[14] Krämer, M., Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compositio Math. 38(1979), no. 2, 129-153.Google Scholar
[15] Nguyêñ, H., Characterizing weakly symmetric spaces as Gelfand pairs. J. Lie Theory 9(1999), no. 2, 285-291.Google Scholar
[16] A., Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20(1956), 47-87.Google Scholar
[17] Z., Shen, Volume comparison and its applications in Riemann-Finsler geometry. Adv. Math. 128(1997), no. 2, 306-328. doi:10.1006/aima.1997.1630Google Scholar
[18] Z., Shen, Differential Geometry of Spray and Finsler Spaces. Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
[19] Z. I., Szabó, Positive definite Berwald spaces. Structure theorems on Berwald spaces. Tensor 35(1981), no. 1, 25-39.Google Scholar
[20] J., Tits, Sur certaine d'espaces homogènes de groupes de Lie. Acad. Roy. Belg. Cl. Sci.Mém. Coll. 29(1955), no. 3.Google Scholar
[21] Wang, H. C., Two-point homogeneous spaces. Ann. of Math. 55(1952), 177-191. doi:10.2307/1969427Google Scholar
[22] J. K., Hale, Weakly symmetric spaces of semisimple Lie groups. Moscow Univ. Math. Bull. 57(2002), no. 2, 37-40 (translation).Google Scholar
[23] J. K., Hale, Weakly symmetric Riemannian manifolds with a reductive isometry group. Sb. Math 195(2004), no. 3-4, 599-614 (translation).Google Scholar
[24] W., Ziller, Weakly symmetric spaces. In: Topics in Geometry. Prog. Nonlinear Differential Equations 20. BirkhÄuser Boston, Boston, 1996, pp. 355-368.Google Scholar