Introduction to Finsler Geometry

  • Shaoqiang Deng
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we will give a brief introduction to Finsler geometry. In Sect. 1.1, we give the definitions of Minkowski spaces and Finsler spaces. Many explicit and interesting examples are presented in this section. Section 1.2 is devoted to introducing the Chern connection and flag curvature. In Sect.1.3, we deduce the formula of the first variation of the arc length and give a detailed account of the exponential map of a Finsler space. We then introduce some results on Jacobi fields and present the Cartan–Hadarmard theorem and the Bonnet–Myers theorem in Sects. 1.4 and 1.5. In Sect. 1.6, we define the notions of Berwald spaces and Landsberg spaces. In Sect. 1.7, we recall Shen’s definition of S-curvature of Finsler spaces. Finally, in Sect.1.8, we collect some results on Finsler spaces of constant flag curvature as well as Einstein–Finsler spaces, and present the Akbar–Zadeh theorem.

Keywords

Manifold Clarification Bonnet 

References

  1. 1.
    Abate, M., Patrizio, G.: Finsler metrics – a global approach. Lecture notes in Math., vol. 1591. Springer, Berlin (1994)Google Scholar
  2. 2.
    Akabar-Zadeh, H.: Sur les espaces de Finsler à courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci. (5) 74, 281–322 (1988)Google Scholar
  3. 3.
    Akhiezer, D.N., Vinberg, E.B.: Weakly symmetric spaces and spherical varieties. Transformation Groups 4, 3–24 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alekseevskii, D.V.: Classification of quaternionic spaces with a transitive solvable group of motions. Izv. Akad. Nauk. SSSR Ser. Mat. 9, 315–362 (1975) (English translation: Math. USSR-Izv 9, 297–339 (1975))Google Scholar
  5. 5.
    Alekseevskii, D.V., Kinmel’fel’d, B.N.: Structure of homogeneous Riemannian manifolds with zero Ricci curvature. Funct. Anal. Appl. 9, 95–102 (1975)Google Scholar
  6. 6.
    Aloff, S., Wallach, N.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–97 (1975)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ambrose, W., Singer, I.M.: A theorem on holonomy. Trans. Am. Math. Soc. 75, 428–443 (1953)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ambrose, W., Singer, I.M.: On homogeneous Riemannian manifolds. Duke Math. J. 25, 647–669 (1958)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    An, H., Deng, S.: Invariant (α, β)-metrics on homogeneous manifolds. Monatsh. Math. 154, 89–102 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler spaces with Applications in Physics and Biology. Kluwer Academic, Dordrecht (1993)MATHGoogle Scholar
  11. 11.
    Bacso, S., Matsumoto, M.: Randers spaces with the h-curvature tensor H dependent on position alone. Publ. Math. Debrecen 57, 185–192 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Baez, J.C.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2002)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Bao, D.: Randers space forms. Periodica Mathematica Hungarica 48, 3–15 (2004)MATHCrossRefGoogle Scholar
  14. 14.
    Bao, D.: On two curvature-driven problems in Finsler geometry. Adv. Pure Math. 48, 19–71 (2007)Google Scholar
  15. 15.
    Bao, D., Chern, S.S.: On a notable connection in Finsler geometry. Houston J. Math. 19, 135–180 (1993)MathSciNetMATHGoogle Scholar
  16. 16.
    Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann-Finsler Geometry. Springer, New York (2000)MATHCrossRefGoogle Scholar
  17. 17.
    Bao, D., Lackey, B.: Randers surfaces whose Laplacians have completely positive symbol. Nonlinear Anal. A 38, 27–40 (1999)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Bao, D., Robles, C.: On Randers spaces of constant flag curvature. Rep. Math. Phys. 51, 9–42 (2003)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Bao, D., Robles, C.: Ricci and flag curvatures in Finsler geometry. In: Bao, D., Bryant, R.L., Chern, S.S., Shen, Z. (eds.) A Sampler of Riemannian-Finsler Geometry, pp. 197–260. Cambridge University Press, London (2004)Google Scholar
  20. 20.
    Bao, D., Shen, Z.: Finsler metrics with constant positive curvature on the sphere S 3. J. Lond. Math. Soc. 66, 453–467 (2002)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Bao, D., Robles, C., Shen, Z.: Zermelo navigation on Riemannian manifolds. J. Differ. Geom. 66, 377–435 (2004)MathSciNetMATHGoogle Scholar
  22. 22.
    Bazaikin, Y.V.: On a certain class of 13-dimensional Riemannian manifolds with positive curvature. Siberian Math. J. 37, 1219–1237 (1996)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bérard Bergery, L.: Les variétés Riemannienes homogènes simplement connexes de dimension impair à courbure strictement positive. J. Math. Pures Appl. 55, 47–68 (1976)Google Scholar
  24. 24.
    Berger, M.: Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Scuola Norm. Sup. Pisa (3) 15, 179–246 (1961)Google Scholar
  25. 25.
    Berndt, J., Vanhecke, L.: Geometry of weakly symmetric spaces. J. Math. Soc. Jpn. 48, 745–760 (1996)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Berndt, J., Kowalski, O., Vanhecke, L.: Geodesics in weakly symmetric spaces. Ann. Glob. Anal. Geom. 15, 153–156 (1997)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Berndt, J., Ricci, F., Vanhecke, L.: Weakly symmetric groups of Heisenberg type. Differ. Geom. Appl. 8, 275–284 (1998)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Besse, A.: Einstein Manifolds. Springer, Berlin (2008)MATHGoogle Scholar
  29. 29.
    Bochner, S., Montgomery, D.: Locally compact groups of differentiable transformations. Ann. Math. 47, 639–653 (1946)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Bogoslovski, Yu.G.: Status and perspectives of theory of local anisotropic space-time. Physics of Nuclei and Particles. Moscow State University, Russia (1997) (in Russian)Google Scholar
  31. 31.
    Borel, A., Some remarks about Lie groups transitive on spheres and tori. Bull. Am. Math. Soc. 55, 580–587 (1940)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Brinzei, N.: Projective relations for m-th root metric spaces. Available at the website: http://arxiv1.library.cornell.edu/abs/0711.4781v2/, preprint. Accessed 2007
  33. 33.
    Brion, M.: Classification des espaces homogènes sphériques. Compositio Math. 63, 189–208 (1989)MathSciNetGoogle Scholar
  34. 34.
    Br\ddot{o}cker, T., tom Dieck, T.: Representations of Compact Lie Groups. Springer, New York (1995)Google Scholar
  35. 35.
    Busemann, H., Phadke, B.: Two theorems on general symmetric spaces. Pac. J. Math. 92, 39–48 (1981)MathSciNetMATHGoogle Scholar
  36. 36.
    Cahen, M., Parker, M.: Pseudo-Riemannian symmetric spaces. Mem. Am. Math. Soc. 24(229) (1980)Google Scholar
  37. 37.
    Cartan, E.: Sur une classe remarquable d’espaces de Riemann. Bull. Soc. Math. France 54, 214–264 (1926)MathSciNetGoogle Scholar
  38. 38.
    Cartan, E.: Sur la détermination d’un système orthogonal complet dans un espace de Riemann symétrique clos. Rend. Circ. Mat. Palermo 53, 217–252 (1929)MATHCrossRefGoogle Scholar
  39. 39.
    Chen, X., Deng, S.: A new proof of a theorem of H. C. Wang. Balkan J. Geom. Appl. 16(2), 25–26 (2011)MathSciNetMATHGoogle Scholar
  40. 40.
    Chen, B.Y., Nagano, T.: Totally geodesic submanifolds of symmetric spaces I. Duke Math. J. 44, 745–755 (1977)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Chen, B.Y., Nagano, T.: Totally geodesic submanifolds of symmetric spaces II. Duke Math. J. 45, 405–425 (1978)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Chern, S.S.: On the Euclidean connections in a Finsler space. Proc. Nat. Acad. Sci. USA 29, 33–37 (1943)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Chern, S.S.: Finsler geometry is just Riemannian geometry without the quadratic restriction. Notices Am. Math. Soc. 43, 959–963 (1996)MathSciNetMATHGoogle Scholar
  44. 44.
    Chern, S.S., Shen, Z.: Riemann-Finsler Geometry. World Scientific, Singapore (2004)MATHGoogle Scholar
  45. 45.
    D’Atri, J.E., Ziller, W.: Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Am. Math. Soc. 18(215) (1979)Google Scholar
  46. 46.
    Deicke, A.: Über die Finsler Räume mit A i = 0. Arch. Math. 4, 45–51 (1953)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Deng, S.: Fixed points of isometries of a Finsler space. Publ. Math. Debrecen 72, 469–474 (2008)MathSciNetMATHGoogle Scholar
  48. 48.
    Deng, S.: The S-curvature of homogeneous Randers spaces. Differ. Geom. Appl. 27, 75–84 (2009)MATHCrossRefGoogle Scholar
  49. 49.
    Deng, S.: An algebraic approach to weakly symmetric Finsler spaces. Can. J. Math. 62, 52–73 (2010)MATHCrossRefGoogle Scholar
  50. 50.
    Deng, S.: Invariant Finsler metrics on polar homogeneous spaces. Pac. J. Math. 247, 47–74 (2010)MATHCrossRefGoogle Scholar
  51. 51.
    Deng, S.: On the classification of weakly symmetric Finsler spaces. Israel J. Math. 181, 29–52 (2011)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Deng, S.: Clifford-Wolf translations of Finsler spaces of negative flag curvature, preprint. arXiv: 1204.1048, 2012/arxiv.orgGoogle Scholar
  53. 53.
    Deng, S.: Finsler metrics and the degree of symmetry of a closed manifold. Indiana U. Math. J. 60, 713–727 (2011)MATHCrossRefGoogle Scholar
  54. 54.
    Deng, S.: On the symmetry of Riemannian manifolds. J. Reine Angew. Math., published online. doi:10.1515/CRELLE.2012.040Google Scholar
  55. 55.
    Deng, S., Hou, Z.: The group of isometries of a Finsler space. Pac. J. Math. 207, 149–155 (2002)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Deng, S., Hou, Z.: Invariant Finsler metrics on homogeneous manifolds. J. Phys. A: Math. Gen. 37, 8245–8253 (2004)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Deng, S., Hou, Z.: Minkowski symmetric Lie algebras and symmetric Berwald spaces. Geometriae Dedicata 113, 95–105 (2005)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Deng, S., Hou, Z.: On locally and globally symmetric Berwald spaces. J. Phys. A: Math. Gen. 38, 1691–1697 (2005)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Deng, S., Hou, Z.: Invariant Finsler metrics on homogeneous manifolds II: Complex structures. J. Phys. A: Math. Gen. 39, 2599–2609 (2006)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Deng, S., Hou, Z.: Invariant Randers metrics on homogeneous Riemannian manifold. J. Phys. A: Math. Gen. 37, 4353–4360 (2004); Corrigendum, J. Phys. A: Math. Gen. 39, 5249–5250 (2006)Google Scholar
  61. 61.
    Deng, S., Hou, Z.: Homogeneous Finsler spaces of negative curvature. J. Geom. Phys. 57, 657–664 (2007)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Deng, S., Hou, Z.: On symmetric Finsler spaces. Israel J. Math. 166, 197–219 (2007)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Deng, S., Hou, Z.: Positive definite Minkowski Lie algebras and bi-invariant Finsler metrics on Lie groups. Geometriae Dedicata 136, 191–201 (2008)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Deng, S., Hou, Z.: Homogeneous Einstein–Randers spaces of negative Ricci curvature. C. R. Math. Acad. Sci. Paris 347, 1169–1172 (2009)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Deng, S., Hou, Z.: Naturally reductive homogeneous Finsler spaces. Manuscripta Math. 131, 215–219 (2010)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Deng, S., Hou, Z.: Weakly symmetric Finsler spaces. Comm. Contemp. Math. 12, 309–323 (2010)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Deng, S., Wang, X.: The S-curvature of homogeneous (α, β)-metrics. Balkan J. Geom. Appl. 15(2), 39–48 (2010)MathSciNetMATHGoogle Scholar
  68. 68.
    Deng, S., Xu, M.: Clifford–Wolf translations of Finsler spaces. Forum Math. (2012), published online, doi:10.1515/forum-2012-0032Google Scholar
  69. 69.
    Deng, S., Xu, M.: Clifford–Wolf translations of homogeneous Randers spheres, preprint. arXiv: 1204.5232, 2012/arxiv.orgGoogle Scholar
  70. 70.
    Eschenburg, J.: New examples of manifolds of positive curvature. Invent. Math. 66, 469–480 (1982)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Foulon, P.: Curvature and global rigidity in Finsler manifolds. Houston J. Math. 28, 263–292 (2002)MathSciNetMATHGoogle Scholar
  72. 72.
    Gorbatsevich, V.V.: Invariant intrinsic Finsler metrics on homogeneous spaces and strong subalgebras of Lie algebras. Siberian Math. J. 49, 36–47 (2008)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Gordon, C.S.: Naturally reductive homogeneous Riemannian manifolds. Can. J. Math. 37, 467–487 (1985)MATHCrossRefGoogle Scholar
  74. 74.
    Gordon, C.S.: Homogeneous Riemannian manifolds whose geodesics are orbits. In: Topics in Geometry, in Memory of Joseph D’Atri, pp. 155–174. Birkhäuser, Basel (1996)Google Scholar
  75. 75.
    Gordan, C.S., Kerr, M.: New homogeneous Einstein metrics of negative Ricci curvature. Geometriae Dedicata 19, 75–101 (2001)Google Scholar
  76. 76.
    Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature and Cohomology, II. Academic, New York (1973)MATHGoogle Scholar
  77. 77.
    Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Invent. Math. 149, 619–664 (2002)MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    Grove, K., Wilking, B., Ziller, W.: Positively curved cohomogeneity one manifolds and 3-Sasaki geometry. J. Differ. Geom. 78, 33–111 (2008)MathSciNetMATHGoogle Scholar
  79. 79.
    Grove, K., Verdiani, L., Ziller, W.: An exotic T 1 S 4 with positive curvature, Geometric and Functional Analysis, 21, 499–524 (2011)MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    Heber, J.: Noncompact homogeneous Einstein manifolds. Invent. Math. 133, 279–252 (1998)MathSciNetMATHGoogle Scholar
  81. 81.
    Heintze, E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Helgason, S.: Totally geodesic spheres in compact symmetric spaces. Math. Ann. 165, 309–317 (1966)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Helgason, S.: Differential Geometry, Lie groups and Symmetric Spaces, 2nd edn. Academic, New York (1978)MATHGoogle Scholar
  84. 84.
    Hsiang, W.Y.: The natural metric on \mathop{ SO} (n + 2) ∕ \mathop{ SO} (n) is the most symmetric. Bull. Am. Math. Soc. 73, 55–58 (1967)MathSciNetMATHCrossRefGoogle Scholar
  85. 85.
    Hsiang, W.C., Hsiang, W.Y.: The degree of symmetry of homotopy spheres. Ann. Math. 89, 52–67 (1969)MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    Hu, Z., Deng, S.: Invariant fourth-root metrics on the Grassmannian manifolds. J. Geom. Phys. 61, 18–25 (2011)MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Hu, Z., Deng, S.: Homogeneous Randers spaces of isotropic S-curvature and positive flag curvature. Math. Z. 270, 989–1009 (2012)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Hu, Z., Deng, S.: On flag curvature of homogeneous Randers spaces. Can. J. Math. (2012), published online, doi:10.4153/CJM-2012-004-6Google Scholar
  89. 89.
    Hu, Z., Deng, S.: Ricci-quadratic homogeneous Randers spaces, preprint. arxiv: 1207.1786/arxiv.orgGoogle Scholar
  90. 90.
    Hu, Z., Deng, S.: Three-dimensional homogeneous Finsler spaces. Math. Nachr. 285, 1243–1254 (2012). doi:10.1002/mana.201100100MATHCrossRefGoogle Scholar
  91. 91.
    Huang, L., Mo, X.: On curvature decreasing property of a class of navigation problems. Publ. Math. Debrecen 71, 991–996 (2007)MathSciNetGoogle Scholar
  92. 92.
    Ichijy\bar{o}, Y.: Finsler spaces modeled on a Minkowski space. J. Math. Kyoto Univ. 16, 639–652 (1976)Google Scholar
  93. 93.
    Ingarden, R.S.: On physical applications of Finsler geometry. Contemp. Math. 196, 213–223 (1996)MathSciNetCrossRefGoogle Scholar
  94. 94.
    Jensen, G.R.: Einstein metrics on principal fibre bundles. J. Differ. Geom. 8, 599–614 (1973)MATHGoogle Scholar
  95. 95.
    Kaplan, A.: Fundamental solution for a class of hypo-elleptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258, 147–153 (1980)MATHCrossRefGoogle Scholar
  96. 96.
    Kath, I., Olbrich, M.: Metric Lie algebras with maximal isotropic centre. Math. Z. 246, 23–53 (2004)MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Kath, I., Olbrich, M.: On the structure of pseudo-Riemannian symmetric spaces. Transformation Groups 14, 847–885 (2009)MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Kim, C.W.: Locally symmetric positively curved Finsler spaces. Arch. Math. (Basel) 88, 378–384 (2007)Google Scholar
  99. 99.
    Kobayashi, S.: Fixed points of isometries. Nagoya Math. J. 13, 63–68 (1958)MathSciNetMATHGoogle Scholar
  100. 100.
    Kobayashi, S.: Homogeneous Riemannian manifolds of negative curvature. Tohoku Math. J. 14, 413–415 (1962)MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    Kobayashi, S.: Transformation Groups in Differential Geometry. Springer, Berlin (1972)MATHCrossRefGoogle Scholar
  102. 102.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1, 1963, vol. 2, 1969. Interscience Publishers, New York (1969)Google Scholar
  103. 103.
    Koszul, J.L.: Exposés sue les espaces homogènes symétriques. Pub. Soc. Math. Sao Paulo 77, (1959)Google Scholar
  104. 104.
    Kowalski, O.: Spaces with volume-preserving symmetries and related classes of Riemannian manifolds. Rend. Sem. Mat. Univ. Politec. Torino, Facicolo Speciale, Settembre, pp. 131–158 (1983)Google Scholar
  105. 105.
    Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Un. Mat. Ital. B 7(5), 189–246 (1991)MathSciNetGoogle Scholar
  106. 106.
    Krämer, M.: Sphärische untergruppen in Kompakten zusammenhängenden Liegruppen. Compositio Math. 38, 129–153 (1979)MathSciNetMATHGoogle Scholar
  107. 107.
    Ku, H., Mann, L., Sicks, J., Su, J.: The degree of symmetry of a product manifold. Trans. Am. Math. Soc. 146, 133–149 (1969)MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    Ku, H., Mann, L., Sicks, J., Su, J.: The degree of symmetry of a homotopy real projective space. Trans. Am. Math. Soc. 161, 51–61 (1971)MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    Kuiper, N.H.: Groups of motions of order \(\frac{1} {2}n(n - 1) + 1\) in Riemannian n-spaces. Indag. Math. 48, 313–318 (1955)Google Scholar
  110. 110.
    Lauret, J.: Homogeneous nilmanifolds attached to representations of compact Lie groups. Manuscripta Math. 99, 287–309 (1999)MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Lauret, J.: Einstein solvmanifolds are standard. Ann. Math. 172, 1859–1877 (2010)MathSciNetMATHGoogle Scholar
  112. 112.
    Lawson, H., Yau, S.T.: Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres. Comment. Math. Helv. 49, 232–244 (1974)MathSciNetMATHCrossRefGoogle Scholar
  113. 113.
    Li, B., Shen, Z.: On projectively flat fourth root metrics. Can. Math. Bull. 55, 138–145 (2012)MathSciNetMATHCrossRefGoogle Scholar
  114. 114.
    Mather, J.: Differentiable invariant. Topology 16, 145–155 (1977)MathSciNetMATHCrossRefGoogle Scholar
  115. 115.
    Matsumoto, M.: Foundations of Finsler Geometry and Special Finsler Spaces. Kaiseisha Press, Japan (1986)MATHGoogle Scholar
  116. 116.
    Matveev, V.S., Troyanov, M.: The Binet–Legender ellipsoid in Finsler geometry. Math. DG arxiv: 1104–1647.v1, preprint. Accessed 2012Google Scholar
  117. 117.
    Mazur, S.: Quelques propriétés characteristiques des espaces Euclidiens. C. R. Acad. Sci. Paris Ser. I 207, 761–764 (1938)Google Scholar
  118. 118.
    Mestag, T.: Relative equilibria of invariant lagrangian systems on a Lie group. In: Differential Geometry Methods in Mechanics and Field Theory, pp. 115–129. Academic, New York (2007)Google Scholar
  119. 119.
    Mikityuk, I.V.: On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Math. Sb. 57, 527–546 (1987)CrossRefGoogle Scholar
  120. 120.
    Milnor, J.: Curvature of left invariant Riemannian metrics on Lie groups. Adv. Math. 21, 293–329 (1976)MathSciNetMATHCrossRefGoogle Scholar
  121. 121.
    Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math. 44, 454–470 (1943)MathSciNetMATHGoogle Scholar
  122. 122.
    Montgomery, D., Yang, C.T.: The existence of a slice. Ann. Math. 65, 108–116 (1957)MathSciNetMATHCrossRefGoogle Scholar
  123. 123.
    Myers, S.B., Steenrod, N.: The group of isometries of a Riemannian manifold. Ann. Math. 40, 400–416 (1939)MathSciNetCrossRefGoogle Scholar
  124. 124.
    Nguyêñ, H.: Characterizing weakly symmetric spaces as Gelfand pairs. J. Lie Theor. 9, 285–291 (1999)MATHGoogle Scholar
  125. 125.
    Obata, M.: On n-dimensional homogeneous spaces of Lie groups of dimension greater than \(\frac{1} {2}n(n - 1)\). J. Math. Soc. Jpn. 7, 371–388 (1955)MathSciNetMATHCrossRefGoogle Scholar
  126. 126.
    Onishchik, A.L.: Transitive compact transformation groups. Math. Sb. 60, 447–485 (1963)Google Scholar
  127. 127.
    Palais, R.S.: A global formulation of the Lie theory of transformation groups. Mem. Am. Math. Soc. 22m (1957)Google Scholar
  128. 128.
    Palais, R.S.: On the differentiability of isometries. Proc. Am. Math. Soc. 8, 805–7807 (1957)MathSciNetMATHCrossRefGoogle Scholar
  129. 129.
    Pavlov, D.G. (ed.): Space-Time Structure. Collected papers, TETRU (2006)Google Scholar
  130. 130.
    Popov, V.L., Vinberg, E.B.: Invariant theory. In: Algebraic Geometry IV. Springer, Berlin (1994)Google Scholar
  131. 131.
    Randers, G.: On an asymmetrical metric in the four-space of general relativity. Phys. Rev. 59, 195–199 (1941)MathSciNetCrossRefGoogle Scholar
  132. 132.
    del Riego, L.: Tenseurs de Weyl d’un spray de directions, Thesis, Université Scientifique et Medicale de Grenoble (1973)Google Scholar
  133. 133.
    Robles, C.: Einstein metrics of Randers type, doctoral dissertation, University of British Colombia (2003)Google Scholar
  134. 134.
    Schultz, R.: Semifree circle actions and the degree of symmetry of homotopy spheres. Am. J. Math. 93, 829–839 (1971)MATHCrossRefGoogle Scholar
  135. 135.
    Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. B. 20, 47–87 (1956)MathSciNetMATHGoogle Scholar
  136. 136.
    Shankar, K.: Isometry groups of homogeneous spaces with positive sectional curvature. Differ. Geom. Appl. 14, 57–78 (2001)MathSciNetMATHCrossRefGoogle Scholar
  137. 137.
    Shen, Z.: Finsler manifolds of constant positive curvature. Contemp. Math. 196, 83–93 (1996)CrossRefGoogle Scholar
  138. 138.
    Shen, Z.: Volume comparison and its applications in Riemann–Finsler geometry. Adv. Math. 128, 306–328 (1997)MathSciNetMATHCrossRefGoogle Scholar
  139. 139.
    Shen, Z.: Differential Geometry of Spray and Finsler Spaces. Kluwer Academic, Dordrecht (2001)MATHGoogle Scholar
  140. 140.
    Shen, Z.: On projectively related Einstein metrics in Riemann–Finsler geometry. Math. Ann. 320, 625–647 (2001)MathSciNetMATHCrossRefGoogle Scholar
  141. 141.
    Shen, Z.: On R-quadratic Finsler spaces. Publ. Math. Debrecen 58, 263–274 (2001)MathSciNetMATHGoogle Scholar
  142. 142.
    Shen, Z.: Two-dimensional Finsler metrics with constant flag curvature. Manuscripta Math. 109, 349–366 (2002)MathSciNetMATHCrossRefGoogle Scholar
  143. 143.
    Shen, Z.: Finsler metrics with K = 0 and S = 0. Can. J. Math. 55, 112–132 (2003)MATHCrossRefGoogle Scholar
  144. 144.
    Shen, Z.: Landsberg curvature, S-curvature and Riemann curvature. In: Bao, D., Bryant, R., Chern, S.S., Shen, Z. (eds.) A Sampler of Finsler Geometry, pp. 303–355, MSRI Publications, no. 50. Cambridge University Press, London (2004)Google Scholar
  145. 145.
    Shen, Z.: Finsler manifolds with non-positive flag curvature and constant S-curvature. Math. Z. 249, 625–639 (2005)MathSciNetMATHCrossRefGoogle Scholar
  146. 146.
    Shen, Z.: Some open problems in Finsler geometry. Available on Z. Shen’s home page at www.math.iupui.edu. Preprint (2009)Google Scholar
  147. 147.
    Shen, Z., Xing, H.: On Randers metrics of isotropic S-curvature. Acta Math. Sinica 24, 789–796 (2008)MathSciNetMATHCrossRefGoogle Scholar
  148. 148.
    Shimada, H.: On Finsler spaces with the metric \(L = {({a}_{{i}_{1}{i}_{2}\cdots {i}_{m}}{y}^{{i}_{1}}{y}^{{i}_{2}}\cdots {y}^{{i}_{m}})}^{ \frac{1} {m} }\). Tensor N.S. 33(3), 365–372 (1979)Google Scholar
  149. 149.
    Simons, J.: On the transitivity of holonomy systems. Ann. Math. 76, 143–171 (1962)MathSciNetCrossRefGoogle Scholar
  150. 150.
    Spiro, A.: Chern’s orthonormal frame bundle of a Finsler space. Houston J. Math. 25, 641–659 (1999)MathSciNetMATHGoogle Scholar
  151. 151.
    Szabó, Z.I.: Generalized spaces with many isometries. Geometriae Dedicata 11, 369–383 (1981)MathSciNetMATHCrossRefGoogle Scholar
  152. 152.
    Szabó, Z.I.: Positive definite Berwald spaces. Tensor N. S. 38, 25–39 (1981)Google Scholar
  153. 153.
    Szabó, Z.I.: Spectral geometry for operator families on Riemannian manifolds. Sympos. Pure Math. 54, 615–665 (1993)Google Scholar
  154. 154.
    Tits, J.: Sur certaine d’espaces homogènes de groupes de Lie. Acad. Roy. Belg. Cl. Sci. Mém. Coll. 29(3) (1955)Google Scholar
  155. 155.
    Tits, J.: Espaces homogènes complexes. Comment. Math. Helv. 37, 111–120 (1963)MathSciNetCrossRefGoogle Scholar
  156. 156.
    Van Danzig, D., Van Der Waerden, B.L.: Über metrische homogene Räume. Abh. Math. Sem. Univ. Hamburg 6, 367–376 (1928)CrossRefGoogle Scholar
  157. 157.
    Varadarajan, V.S.: Lie Groups, Lie Algebras and Their Representations. Springer, New York (1984)MATHGoogle Scholar
  158. 158.
    Verdianni, L., Ziller, W.: Positively curved homogeneous metrics on spheres. Math. Z. 261, 473–488 (2009)MathSciNetCrossRefGoogle Scholar
  159. 159.
    Wallach, N.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96, 277–295 (1972)MathSciNetMATHCrossRefGoogle Scholar
  160. 160.
    Walter, T.H.: Einstein metrics on solvable groups. Math Z. 206, 457–471 (1991)MathSciNetCrossRefGoogle Scholar
  161. 161.
    Wang, H., Deng, S.: Some homogeneous Einstein–Randers spaces. Nonlinear Anal. 72, 4407–4414 (2010)MathSciNetMATHCrossRefGoogle Scholar
  162. 162.
    Wang, H., Deng, S.: Left invariant Einstein-Randers metrics on compact Lie groups. Can. Math. Bull. (2011), published online, doi:10.4153/CMB-2011-145-6Google Scholar
  163. 163.
    Wang, H.C.: Finsler spaces with completely integrable equations of Killing. J. Lond. Math. Soc. 22, 5–9 (1947)MATHCrossRefGoogle Scholar
  164. 164.
    Wang, H.C.: Two point homogeneous spaces. Ann. Math. 55, 177–191 (1952)MATHGoogle Scholar
  165. 165.
    Wang, M., Ziller, W.: On normal homogeneous Einstein manifolds. Ann. Sci. Ec. Norm. Sup. 18, 563–633 (1985)MathSciNetMATHGoogle Scholar
  166. 166.
    Wang, M., Ziller, W.: Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84, 177–194 (1986)MathSciNetMATHCrossRefGoogle Scholar
  167. 167.
    Wang, M., Ziller, W.: On isotropy irreducible Riemannian manifolds. Acta Math. 116, 223–261 (1991)MathSciNetCrossRefGoogle Scholar
  168. 168.
    Wang, H., Huang, L., Deng, S.: Homogeneous Einstein-Randers metrics on spheres. Nonlinear Anal. 74, 6295–6301 (2011)MathSciNetMATHCrossRefGoogle Scholar
  169. 169.
    Wilking, B.: The normal homogeneous space \mathop{ SU} (3) ×\mathop{ SO} (3) ∕ \mathop{ U}  ∗ (2) has positive sectional curvature. Proc. Am. Math. Soc. 127, 1191–1194 (1999)MathSciNetMATHCrossRefGoogle Scholar
  170. 170.
    Wilson, E.N.: Isometry groups on homogeneous nilmanifolds. Geometriae Dedicata 12, 337–346 (1982)MathSciNetMATHCrossRefGoogle Scholar
  171. 171.
    Wolf, J.A.: Sur la classification des variétés riemanniennes homogènes à courbure constante. C. R. Acad. Sci. Paris Ser. I 250, 3443–3445 (1960)MATHGoogle Scholar
  172. 172.
    Wolf, J.A.: Locally symmetric homogeneous spaces. Comment. Math. Helv. 37, 65–101 (1962–1963)Google Scholar
  173. 173.
    Wolf, J.A.: Curvature in nilpotent Lie groups. Proc. Am. Math. Soc. 15, 271–274 (1964)MATHCrossRefGoogle Scholar
  174. 174.
    Wolf, J.A.: Homogeneity and bounded isometries in manifolds of negative curvature. Illinois J. Math. 8, 14–18 (1964)MathSciNetMATHGoogle Scholar
  175. 175.
    Wolf, J.A.: The geometry and structure of isotropic irreducible homogeneous spaces. Acta Math. 20, 59–148 (1968); Correction, Acta Math. 152, 141–142 (1984)Google Scholar
  176. 176.
    Wolf, J.A.: Harmonic Analysis on Commutative Spaces. Amer. Math. Soc. vol. 142 (2007)Google Scholar
  177. 177.
    Wolf, J.A.: Spaces of Constant Curvature, 6th edn. Amer. Math. Soc. Chelsea. American Mathematical Society, Providence (2011)MATHGoogle Scholar
  178. 178.
    Xu, B.: The degree of symmetry of certain compact smooth manifolds. J. Math. Soc. Jpn. 55, 727–737 (2003)MATHCrossRefGoogle Scholar
  179. 179.
    Yakimova, O.: Weakly symmetric spaces of semisimple Lie groups. Moscow Univ. Math. Bull. 57(2), 37–40 (2002)MathSciNetGoogle Scholar
  180. 180.
    Yakimova, O.: Weakly symmetric Riemannian manifolds with a reductive isometry group. Sb. Math. 195(3–4), 599–614 (2004)MathSciNetMATHCrossRefGoogle Scholar
  181. 181.
    Yano, K.: On n-dimensional Riemannian manifolds admitting a group of motions of order \(\frac{1} {2}n(n - 1) + 1\). Trans. Am. Math. Soc. 74, 260–279 (1953)MATHGoogle Scholar
  182. 182.
    Ziller, W.: Homogeneous Einstein metrics on spheres. Math. Ann. 259, 351–358 (1982)MathSciNetMATHCrossRefGoogle Scholar
  183. 183.
    Ziller, W.: Weakly symmetric spaces. In: Topics in Geometry, in memory of Joseph D’Atri. Progress in Nonlinear Diff. Eqs. 20, pp. 355–368. Birkhäuser, Basel (1996)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Shaoqiang Deng
    • 1
  1. 1.School of Mathematical SciencesNankai UniversityTianjinPeople’s Republic of China

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