Journal of Mathematical Sciences

, Volume 146, Issue 6, pp 6313–6390 | Cite as

Geometry of homogeneous Riemannian manifolds

  • Yu. G. Nikonorov
  • E. D. Rodionov
  • V. V. Slavskii


Symmetric Space Sectional Curvature Homogeneous Space Ricci Curvature Einstein Metrics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    U. Abresh and W. T. Meyer, “Injectivity radius estimates and sphere theorems,” in: Comparison Geometry (K. Grove et al., eds.), Cambridge (1997), pp. 1–47.Google Scholar
  2. 2.
    M. A. Akivis and V. V. Goldberg, “On geometry of hypersurfaces of a pseudoconformal space of Lorentzian signature,” J. Geom. Phys., 26, 112–126 (1998).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    D. V. Aleekseevsky, “Conjugacy of polar decompositions of Lie groups,” Mat. Sb., 84, 14–26 (1971).MathSciNetGoogle Scholar
  4. 4.
    D. V. Aleekseevsky, “Classification of quaternion spaces with transitive solvable motion group,” Izv. Akad. Nauk SSSR, Ser. Mat., 39, No. 2, 315–362 (1975).MathSciNetGoogle Scholar
  5. 5.
    D. V. Aleekseevsky, “Homogeneous Riemannian spaces of negative curvature,” Mat. Sb., 96, 93–117 (1975).MathSciNetGoogle Scholar
  6. 6.
    D. V. Alekseevsky, “Homogeneous Einstein metrics,” in: Differential Geometry and Its Application. Proc. of Brno Conf., Univ. of J. E. Purkyne (1987), pp. 1–12.Google Scholar
  7. 7.
    D. Alekseevsky and A. Arvanitoyeorgos, “Metrics with homogeneous geodesics on flag manifolds,” Commun. Math. Univ. Carol., 43, No. 2, pp. 189–199.Google Scholar
  8. 8.
    D. Alekseevsky, I. Dotti, and S. Ferraris, “Homogeneous Ricci positive 5-manifolds,” Pac. J. Math., 175, 1–12 (1996).MathSciNetGoogle Scholar
  9. 9.
    D. Alekseevsky and B. N. Kimelfeld, “Structure of homogeneous Riemannian spaces with zero Ricci curvature,” Funct. Anal. Appl., 9, 27–102 (1975).CrossRefGoogle Scholar
  10. 10.
    D. V. Aleekseevsky and B. Kimelfeld, “Classification of homogeneous conformally flat Riemannian manifolds,” Mat. Zametki, 24 (1978).Google Scholar
  11. 11.
    S. Aloff and N. Wallach, “An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures,” Bull. Amer. Math. Soc., 81, 93–97 (1975).zbMATHMathSciNetGoogle Scholar
  12. 12.
    W. Ambrose and I. M. Singer, “On homogeneous Riemannian manifolds,” Duke Math. J., 25, 647–669 (1958).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    B. N. Apanasov, “Kobayashi conformal metric on manifolds Chern-Simons and η-invariants,” Int. J. Math., 2, No. 4, 361–382 (1991).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Arvanitoyergos, “New invariant Einstein metrics on generalized flag manifolds,” Trans. Amer. Math. Soc., 337, 981–995 (1993).MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Arvanitoyergos, “SO(n)-invariant Einsten metrics on Stiefel manifolds,” in: Proc. Conf. Diff. Geom. Appl. Aug.-Sept. 1995, Brno (1995), pp. 1–5.Google Scholar
  16. 16.
    A. Arvanitoyergos, “Einstein equations for invariant metrics on generalized flag manifolds and inner automorphisms,” Balkan J. Geom. Appl., 1, 17–22 (1996).Google Scholar
  17. 17.
    R. Azencott and E. Wilson, “Homogeneous manifolds with negative curvature, I,” Trans. Amer. Math. Soc., 215, 323–362 (1976).zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    R. Azencott and E. Wilson, “Homogeneous manifolds with negative curvature, II,” Mem. Amer. Math. Soc., 178 (1976)Google Scholar
  19. 19.
    A. Back and W. Y. Hsiang, “Equivariant geometry and Kervaire spheres,” Trans. Amer. Math. Soc., 304, 207–227 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ya. V. Bazaikin, “On a certain family of closed 13-dimensional Riemannian manifolds of positive curvature,” Sib. Mat. Zh., 37, 1068–1085 (1996).zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Berger, “Les varietes riemanniennes homogenes normales a courbure strictement positive,” Ann. Sc. Norm. Sup. Pisa, 15, 179–246 (1961).zbMATHGoogle Scholar
  22. 22.
    L. B. Bergery, “Les varietes riemanniens invariantes homogenes simplement connexes de dimension impaire a courbure strictement positive,” J. Math. Pur. Appl. IX. Ser., 55, No. 1, 47–68 (1976).zbMATHGoogle Scholar
  23. 23.
    L. B. Bergery, “Sur la courbure des metrigues riemanniens invariantes des groupes de Lie et des espaces homogenes,” Ann. Sci. Ecole Norm. Super., 4, No. 4, 543–576 (1978).Google Scholar
  24. 24.
    L. B. Bergery, “Homogeneous Riemannian spaces of dimension 4,” in: Gèomè trie Riemannienne en dimension 4. Sèminaire Arthur Besse, Cedic, Paris (1981).Google Scholar
  25. 25.
    V. N. Berestovskii, “Homogeneous Riemannian manifolds of positive Ricci curvature,” Mat. Zametki, 58, No. 3, 334–340 (1995).MathSciNetGoogle Scholar
  26. 26.
    V. N. Berestovskii and D. E. Volper, “A class of U(n)-invariant Riemannian metrics on manifolds diffeomorphic to the odd-dimensional spheres,” Sib. Mat. Zh., 34, No. 4, 612–619 (1993).MathSciNetCrossRefGoogle Scholar
  27. 27.
    J. Berndt, O. Kowalski, and L. Vanhecke, “Geodesics in weakly symmetric spaces,” Ann. Global Anal. Geom., 15, 153–156 (1997).zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    A. Besse, Einstein Manifolds, Vols. 1, 2 [Russian translation], Mir, Moscow (1990).zbMATHGoogle Scholar
  29. 29.
    A. L. Besse, Manifolds All of Whose Geodesics are Closed [Russian translation], Mir, Moscow (1981).Google Scholar
  30. 30.
    S. Bochner, “Vectors fields and Ricci curvature,” Bull. Ann. Math. Soc., 52, 776–797 (1946).zbMATHMathSciNetGoogle Scholar
  31. 31.
    A. Borel, “Some remarks about Lie groups transitive on spheres and tori,” Bull. Amer. Math. Soc., 55, 580–587 (1949).zbMATHMathSciNetGoogle Scholar
  32. 32.
    A. Borel, “Le plant projectif des octaves et les spheres comme espaces homogenes,” C. R. Acad. Sci. Paris, 230, 1378–1380 (1950).zbMATHMathSciNetGoogle Scholar
  33. 33.
    A. A. Borisenko, Intrinsic and Extrinsic Geometry of Multi-Dimensional Submanifolds [in Russian], Moscow (2003).Google Scholar
  34. 34.
    A. A. Borisenko, “On cylindrical many-dimensional surfaces in the Lobachevskii space,” Ukr. Geom. Sb., 33, 18–27 (1990).zbMATHMathSciNetGoogle Scholar
  35. 35.
    A. A. Borisenko, “Extrinsic geometry of strongly parabolic many-dimensional submanifolds,” Usp. Mat. Nauk, 52, No. 6, 3–52 (1997).MathSciNetGoogle Scholar
  36. 36.
    A. A. Borisenko, Exterior geometry of parabolic and saddle many-dimensional submanifolds,” Usp. Mat. Nauk, 53, No. 6, 3–52 (1998).MathSciNetGoogle Scholar
  37. 37.
    J. P. Bourguignon and H. Karcher, “Curvature operators: Pinching estimates and geometric examples,” Ann. Sci. Ecole Norm. Super., 11, 71–92 (1978).zbMATHMathSciNetGoogle Scholar
  38. 38.
    C. Boyer, K. Galicki, and B. Mann, “The geometry and topology of 3-Sasakian manifolds,” J. Reine Angew. Math., 455, 183–220 (1994).zbMATHMathSciNetGoogle Scholar
  39. 39.
    C Böhm, Homogeneous Einstein metrics and simplicial complexes, Preprint (2003).Google Scholar
  40. 40.
    C. Böhm, Non-existence of homogeneous Einstein metrics, Preprint (2003).Google Scholar
  41. 41.
    C. Böhm and M. Kerr, Low-dimensional homogeneous Einstein manifolds, Preprint (2002).Google Scholar
  42. 42.
    C. Böhm, M. Wang, and W. Ziller, “A variational approach for compact homogeneous Einstein manifolds” (in press).Google Scholar
  43. 43.
    G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York-London (1972).zbMATHGoogle Scholar
  44. 44.
    É. Cartan, “Sur les domaines bornès homogènes de l’space de n variables complexes,” Abh. Math. Sem. Univ. Hamburg, 11, 116–162 (1935).zbMATHGoogle Scholar
  45. 45.
    L. Castellani and L. J. Romans, “N = 3 and N = 1_supersymmetry in a new class of solutions for D = 11 supergravity,” Nucl. Phys. B, 241, 683–701 1984).MathSciNetCrossRefGoogle Scholar
  46. 46.
    L. Castellani, R. D’Auria, and P. Fre, “SU(3)ΘSU(2)ÈU(1) for D = 11 supergravity,” Nucl. Phys. B, 239, 610–652 (1984).MathSciNetCrossRefGoogle Scholar
  47. 47.
    L. Castellani, L. J. Romans, and N. P. Warner, “A classification of compactifying solutions for d = 11 supergravity,” Nucl. Phys. B, 241, 429–462 (1984).MathSciNetCrossRefGoogle Scholar
  48. 48.
    I. Chavel, “Isotropic Jacobi fields and Jacobi’s equation on Riemannian homogeneous spaces,” Comment. Math. Helv., 42, 237–248 (1967).zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    C. D. Collinson, “A comment on the integrability conditions of the conformal Killing equation,” Gen. Relativ. Gravit., 21, No. 9, 979–980 (1989).zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    E. Cortés, “Alekseevskian spaces,” Differ. Geom. Appl., 6, 129–168 (1996).zbMATHCrossRefGoogle Scholar
  51. 51.
    J. E. D’Atri and H. K. Nickerson, “Geodesic symmetrics in spaces with special curvature tensor,” J. Differ. Geom., 9, 251–262 (1974).zbMATHMathSciNetGoogle Scholar
  52. 52.
    J. E. D’Atri and W. Ziller, “Naturally reductive metrics and Einstein metrics on compact Lie groups,” Mem. Amer. Math. Soc., 18, No. 215, 1–72 (1979).MathSciNetGoogle Scholar
  53. 53.
    R. D’Auria, P. Fre, and P. van Nieuwenhuisen, “N = 2 matter coupled supergravity from compactification on a coset G/H possessing an additional Killing vector,” Phys. Lett. B, 136, 347–353 (1984).MathSciNetCrossRefGoogle Scholar
  54. 54.
    E. Deloff, Naturally reductive metrics and metrics with volume preserving geodesic symmetries on NC-algebras, Thesis, Rutgers, New Brunswick (1979).Google Scholar
  55. 55.
    I. Dotti Miatello, “Ricci curvature of left-invariant metrics on solvable unimodular Lie groups,” Math. Z., 180, 257–263 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    I. Dotti Miatello, “Transitive group actions and Ricci curvature properties,” Michigan Math. J., 35, 427–434 (1988).zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    M. J. Duff, B. E. W. Nilsson, and C. N. Pope, “Kaluza-Klein supergravity,” Phys. Rep., 130, 1–142 (1986).MathSciNetCrossRefGoogle Scholar
  58. 58.
    Z. Dusek, “Structure of geodesics in a 13-dimensional group of Heisenberg type,” in: Proc. Coll. Differ. Geom. in Debrecen (2001), pp. 95–103.Google Scholar
  59. 59.
    Z. Dusek, Explicit geodesic graphs on some H-type groups, Preprint.Google Scholar
  60. 60.
    E. B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras, Mat. Sb., 30, No. 2, 349–462 (1952).MathSciNetGoogle Scholar
  61. 61.
    E. B. Dynkin, “Maximal subgroups of the classical groups,” Tr. Mosk. Mat. Obshch., 1, 39–166 (1952).MathSciNetzbMATHGoogle Scholar
  62. 62.
    H. I. Eliasson, “Die Krummung des Raumes Sp(2)/ SU(2) von Berger,” Math. Ann., 164, 317–327 (1966).zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Y.-H. Eschenburg, “New examples of manifolds with strictly positive curvature,” Invent. Math., 66, 469–480 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    S. G. Gindikin, I. I. Pyatetskii-Shapiro, and E. B. Vinberg, “On classification and canonical realization of bounded homogeneous domains,” Tr. Mosk. Mat. Obshch., 12, 359–388 (1963).Google Scholar
  65. 65.
    S. G. Gindikin, I. I. Piatetskii-Shapiro, and E. B. Vinberg, “Homogeneous Kähler manifolds,” in: Homogeneous Bounded Domains (C.I.M.E., 3 Ciclo, Urbino, 1967), Edizioni Cremoneze, Roma (1968), pp. 3–87.Google Scholar
  66. 66.
    V. M. Gol’dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).Google Scholar
  67. 67.
    C. Gordon, “Homogeneous manifolds whose geodesics are orbits,” in: Topics in Geometry. In Memory of J. D’Atri, Birkhauser, Boston (1996), pp. 155–174.Google Scholar
  68. 68.
    C. S. Gordon and M. Kerr, “New homogeneous metrics of negative Ricci curvature,” Ann. Global Anal. Geom., 19, 1–27 (2001).MathSciNetCrossRefGoogle Scholar
  69. 69.
    J. Heber, “Noncompact homogeneous Einstein spaces,” Invent. Math., 133, 279–352 (1998).zbMATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    E. Heintze, “The curvature of SU(5)/ Sp(2) × S 1,” Invent. Math., 13, 205–212 (1971).zbMATHMathSciNetCrossRefGoogle Scholar
  71. 71.
    E. Heintze, “Riemannsche Solvmannigfaltigkeiten,” Geom. Dedic., 1, 141–147 (1973).zbMATHMathSciNetGoogle Scholar
  72. 72.
    E. Heintze, “On homogeneous manifolds of negative curvature,” Math. Ann., 211, 23–34 (1974).zbMATHMathSciNetCrossRefGoogle Scholar
  73. 73.
    S. Helgason, Differential Geometry and Symmetric Spaces [Russian translation], Mir, Moscow (1964).zbMATHGoogle Scholar
  74. 74.
    H.-M. Huang, “Some remarks on the pinching problems,” Bull. Inst. Math. Acad. Sin., 9, 321–340 (1981).zbMATHGoogle Scholar
  75. 75.
    G. R. Jensen, “Homogeneous Einstein spaces of dimension 4,” J. Differ. Geom., 3, 309–349 (1969).zbMATHGoogle Scholar
  76. 76.
    G. R. Jensen, “The scalar curvature of left-invariant Riemannian metrics,” Indiana Univ. Math. J., 20, 1125–1143 (1971).MathSciNetCrossRefGoogle Scholar
  77. 77.
    G. R. Jensen, “Einstein metrics on principal fibre bundles,” J. Differ. Geom., 8, 599–614 (1973).zbMATHGoogle Scholar
  78. 78.
    B. E. Kantor and S. A. Frangulov, “On isometric immersion of two-dimensional Riemannian manifolds in the pseudo-Euclidean space,” Mat. Zametki, 36, No. 3, 447–455 (1984).MathSciNetGoogle Scholar
  79. 79.
    A. Kaplan, “On the geometry of groups of Heisenberg type,” Bull. London Math. Soc., 15, 35–42 (1983).zbMATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    M. Kerr, “Some new homogeneous Einstein metrics on symmetric spaces,” Trans. Amer. Math. Soc., 348, 153–171 (1996).zbMATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    M. Kerr, “New examples of homogeneous Einstein metrics,” Michigan J. Math., 45, 115–134 (1998).zbMATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    M. Kerr, A deformation of quaternionic hyperbolic space, Preprint (2002).Google Scholar
  83. 83.
    M. Kimura, “Homogeneous Einstein metrics on certain Kähler C-spaces,” Adv. Stud. Pure Math., 18, 303–320 (1990).Google Scholar
  84. 84.
    S. Klaus, Einfach-zusammenhängenge Kompakte Homogene Räume bis zur Dimension Neun, Diplomarbeit am Fachbereich Matthematik, Johannes Gutenberg Universität (1988).Google Scholar
  85. 85.
    W. Klingenberg, Lectures on Closed Geodesics [Russan translation], Mir, Moscow (1982).zbMATHGoogle Scholar
  86. 86.
    B. N. Kimelfeld, “Homogeneous domains on the conformal sphere,” Mat. Zametki, 8, No. 3 (1970).Google Scholar
  87. 87.
    S. Kobayashi, “Topology of positively pinched Käler manifolds,” Tohoku Math. J., 15, 121–139 (1963).zbMATHMathSciNetGoogle Scholar
  88. 88.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1, 2, Interscience Publ., New York (1963, 1969).zbMATHGoogle Scholar
  89. 89.
    B. Kostant, Holonomy and Lie algebra of motions in Riemannian manifolds,” Trans. Amer. Math. Soc., 80, 520–542 (1955).MathSciNetCrossRefGoogle Scholar
  90. 90.
    B. Kostant, “On differential geometry and homogeneous spaces, II,” Proc. Natl. Acad. Sci., 42, 354–357 (1956).zbMATHCrossRefGoogle Scholar
  91. 91.
    O. Kowalski, “Counter-example to the ’second Singer’s theorem,’” Ann. Global Anal. Geom., 8, No. 2, 211–214 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  92. 92.
    O. Kowalski and S. Nikcević, “Eigenvalues of locally homogeneous riemannian 3-manifolds,” Geom. Dedic., 62, 65–72 (1996).zbMATHCrossRefGoogle Scholar
  93. 93.
    O. Kowalski and S. Nikcevi’c, “On geodesic graphs of Riemannian g.o. spaces,” Arch. Math., 73, 223–234 (1999).zbMATHMathSciNetCrossRefGoogle Scholar
  94. 94.
    O. Kowalski, S. Nikcević, and Z. Vlásek, “Homogeneous geodesics in homogeneous Riemannian manifolds—examples,” in: Geometry and Topology of Submanifolds, X, World Scientific Publ., River Edge, New Jersey (2000), pp. 104–112.Google Scholar
  95. 95.
    O. Kowalski and J. Szenthe, “On the existence of homogeneous geodesics in homogeneous Riemannian manifolds,” Geom. Dedic., 81, 209–214 (2000); correction: Geom. Dedic., 84, 331–332 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    O. Kowalski and L. Vanhecke, “Riemannian manifolds with homogeneous geodesics,” Boll. Unione Mat. Ital. VII. Ser. B, 5, No. 1, 189–246 (1991).zbMATHMathSciNetGoogle Scholar
  97. 97.
    O. Kowalski and Z. Vlásek, “Homogeneous Einstein metrics on Aloff-Wallach spaces,” Differ. Geom. Appl., 3, 157–167 (1993).zbMATHCrossRefGoogle Scholar
  98. 98.
    M. Krämer, “Eine Klassifikation bestimmter Untergruppen kompakter zusammenhängender Liegruppen,” Commun. Alg., 3, 691–737 (1975).zbMATHCrossRefGoogle Scholar
  99. 99.
    M. Kreck and S. Stolz, “A diffeomorphism classification on 7-dimensional Einstein manifolds with SU(3)Θ SU(2)Θ U(1) symmetry,” Ann. Math., 127, 373–388 (1988).MathSciNetCrossRefGoogle Scholar
  100. 100.
    M. Kreck and S. Stolz, “Some nondiffeomorphic homeomorphic homogeneous 7-manifolds with positive sectional curvature,” J. Differ. Geom., 33, 465–486 (1991).MathSciNetzbMATHGoogle Scholar
  101. 101.
    N. H. Kuiper, “On conformally-flat spaces in the large,” Ann. Math. (2), 50, 916–924 (1949).MathSciNetCrossRefGoogle Scholar
  102. 102.
    N. H. Kuiper, “On compact conformally Euclidean spaces of dimension (2),” Ann. Math. (2), 52, 478–490 (1950).MathSciNetCrossRefGoogle Scholar
  103. 103.
    E. Lacomba, “Mechanical systems with symmetry on homogeneous spaces,” Trans. Amer. Math. Soc., 185, 477–491 (1974).MathSciNetCrossRefGoogle Scholar
  104. 104.
    G. Lastaria Federico and F. Tricerri, “Curvature-orbits and locally homogeneous Riemannian manifolds,” Ann. Mat. Pura Appl., IV. Ser., 165, 121–131 (1993).zbMATHCrossRefGoogle Scholar
  105. 105.
    J. Lauret, “Ricci soliton homogeneous nilmanifolds,” Math. Ann., 319, 715–733 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  106. 106.
    J. Lauret, “Standard Einstein solvmanifolds as critical points,” Quart. J. Math., 52, 463–470 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  107. 107.
    J. Lauret, “Finding Einstein solvmanifolds by a variational method,” Math. Z., 241, 83–99 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  108. 108.
    M. L. Leite and I. D. Miatello, “Metrics of negative Ricci curvature on SL(n, ℝ), n ≥ 3,” J. Differ. Geom., 17, 635–641 (1982).zbMATHGoogle Scholar
  109. 109.
    A. M. Loshmankov, Yu. G. Nikonorov, and E. V. Firsov, “Invariant Einstein metrics on tri-locally symmetric spaces,” Mat. Tr., 6, No. 2, 80–101 (2003).MathSciNetGoogle Scholar
  110. 110.
    O. V. Manturov, “Homogeneous nonsymmetric Riemannian spaces with irreducible rotation group,” Dokl. Akad. Nauk SSSR, 141, 792–795 (1961).MathSciNetGoogle Scholar
  111. 111.
    O. V. Manturov, “Riemannian spaces with orthogonal and symplectic motion groups and with irreducible rotation group,” Dokl. Akad. Nauk SSSR, 141, 1034–1037 (1961).MathSciNetGoogle Scholar
  112. 112.
    O. V. Manturov, “Homogeneous Riemannian manifolds with irreducible isotropy group,” Tr. Semin. Vekt. Tenzor. Anal., 13, 68–145 (1966).MathSciNetGoogle Scholar
  113. 113.
    Y. McCleary and W. Ziller, “On the free loop space of homogeneous spaces,” Amer. J. Math., 109, 765–781 (1987); correction: Amer. J. Math., 113, 375-377 (1991).zbMATHMathSciNetCrossRefGoogle Scholar
  114. 114.
    M. V. Meshcheryakov, “Several remarks on Hamiltonian flows on homogeneous spaces,” Usp. Mat. Nauk, 40, No. 3, 215–216 (1985).MathSciNetGoogle Scholar
  115. 115.
    J. Milnor, “Curvature of left invariant metrics on Lie groups,” Adv. Math., 21, 293–329 (1976).zbMATHMathSciNetCrossRefGoogle Scholar
  116. 116.
    D. Montgomery and H. Samelson, “Transformation groups of spheres,” Ann. Math., 44, 454–470 (1943).MathSciNetCrossRefGoogle Scholar
  117. 117.
    D. Montgomery and H. Samelson, “Groups transitive on the n-dimensional torus,” Bull. Amer. Math. Soc., 49, 455–456 (1943).zbMATHMathSciNetGoogle Scholar
  118. 118.
    S. B. Myers, “Riemannian manifolds with positive mean curvature,” Duke Math. J., 8, 401–404 (1941).zbMATHMathSciNetCrossRefGoogle Scholar
  119. 119.
    S. B. Myers and N. Steenrod, “The group of isometries of Riemannian manifolds,” Ann. Math., 40, 400–416 (1939).MathSciNetCrossRefGoogle Scholar
  120. 120.
    K. Nakajima, “On j-algebras and homogeneous Kähler manifolds,” Hokkaido Math. J., 15, 1–20 (1986).zbMATHMathSciNetGoogle Scholar
  121. 121.
    E. V. Nikitenko and Yu. G. Nikonorov, Six-dimensional Einstein solvmanifolds, Preprint.Google Scholar
  122. 122.
    I. G. Nikolaev, On the smoothness of the metric of spaces with curvature two-sided bounded in the A. D. Aleksandrov sense,” Sib. Mat. Zh., 24, No. 2, 114–132 (1983).MathSciNetGoogle Scholar
  123. 123.
    Yu. G. Nikonorov, “Scalar curvature functional and homogeneous Einstein metrics on Lie groups,” Sib. Mat. Zh., 39, No. 3, 583–589 (1998).zbMATHMathSciNetGoogle Scholar
  124. 124.
    Yu. G. Nikonorov, “On a certain class of compact homogeneous Einstein manifolds,” Sib. Mat. Zh., 41, No. 1, 200–205 (2000).zbMATHMathSciNetGoogle Scholar
  125. 125.
    Yu. G. Nikonorov, “On the Ricci curvature of homogeneous metrics on noncompact homogeneous spaces,” Sib. Mat. Zh., 41, No. 2, 421–429 (2000).zbMATHMathSciNetGoogle Scholar
  126. 126.
    Yu. G. Nikonorov, “Compact seven-dimensional homogeneous Einstein manifolds,” Dokl. Ross. Akad. Nauk, 372, No. 5, 589–592 (2000).MathSciNetGoogle Scholar
  127. 127.
    Yu. G. Nikonorov, “Classification of invariant Einstein metrics on the Aloff-Wallach spaces,” in: Proc. Conf. “Geometry and Applications” Dedicated to the 70th Birthday of V. A. Toponogov, March 13–16, 2000 [in Russian], Inst. Math. Sib. Department Russ. Acad. Sci., Novosibirsk (2000), pp. 128–145.Google Scholar
  128. 128.
    Yu. G. Nikonorov, “New series of Einstein homogeneous metrics,” Differ. Geom. Appl., 12, 25–34 (2000).zbMATHMathSciNetCrossRefGoogle Scholar
  129. 129.
    Yu. G. Nikonorov, “Invariant Einstein metrics on the Ledger-Obata spaces,” Algebra Analiz, 14, No. 3, 169–185 (2002).zbMATHMathSciNetGoogle Scholar
  130. 130.
    Yu. G. Nikonorov, “Algebraic structure of the standard homogeneous Einstein manifolds,” Mat. Tr., 3, No. 1, 119–143 (2002).MathSciNetGoogle Scholar
  131. 131.
    Yu. G. Nikonorov, “Five-dimensional Einstein solvmanifolds,” in: Works on Geometry and Analysis [in Russian], Inst. Math. Sib. Department Russ. Acad. Sci., Novosibirsk (2003), pp. 343–367.Google Scholar
  132. 132.
    Yu. G. Nikonorov, “Compact homogeneous Einstein 7-manifolds,” Geom. Dedic. (to appear).Google Scholar
  133. 133.
    Yu. G. Nikonorov, “Noncompact homogeneous Einstein 5-manifolds,” Geom. Dedic. (to appear).Google Scholar
  134. 134.
    Yu. Nikonorov and E. Rodionov, “Standard homogeneous Einstein manifolds and Diophantine equations,” Arch. Math. (Brno), 32, 123–136 (1996).zbMATHMathSciNetGoogle Scholar
  135. 135.
    Yu. G. Nikonorov and E. D. Rodionov, “Compact six-dimensional homogeneous Einstein manifolds,” Dokl. Ross. Akad. Nauk, 366, No. 5, 599–601 (1999).zbMATHMathSciNetGoogle Scholar
  136. 136.
    Yu. G. Nikonorov and E. D. Rodionov, “Compact homogeneous Einstein 6-manifolds,” Differ. Geom. Appl., 19, 369–378 (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  137. 137.
    Yu. G. Nikonorov, E. D. Rodionov, and V. V. Slavskii, “Geometry of homogeneous Riemannian manifolds,” in: Treatise on Geometry and Analysis [in Russian], Inst. Math. Sib. Department of Russ. Acad. Sci., Novosibirsk (2003), pp. 162–189.Google Scholar
  138. 138.
    L. A. Onishchik, “Inclusion relations between the transitive compact transformation groups,” Tr. Mosk. Mat. Obshch., 11, 199–242 (1962).zbMATHGoogle Scholar
  139. 139.
    L. A. Onishchik, “Transitive compact transformation groups,” Mat. Sb., 60, 447–485 (1963).MathSciNetGoogle Scholar
  140. 140.
    J. M. Overduin and C. S. Wesson, “Kaluza-Klein gravity,” Phys. Rep., 283, 303–378 (1997).MathSciNetCrossRefGoogle Scholar
  141. 141.
    D. Page and C. Pope, “New squashed solutions of d = 11 supergravity,” Phys. Lett. B, 147, 55–60 (1984).MathSciNetCrossRefGoogle Scholar
  142. 142.
    J. Park, “Einstein normal homogeneous Riemannian manifold,” Proc. Jpn. Acad., 72, 197–198 (1996).zbMATHGoogle Scholar
  143. 143.
    J. Park and Y. Sakane, “Invariant Einstein metrics on certain homogeneous spaces,” Tokyo J. Math., 20, 51–61 (1997).MathSciNetCrossRefGoogle Scholar
  144. 144.
    T. Puttmann, Optimal pinching constants of odd-dimensional homogeneous spaces, Ph.D. Thesis, Bochum Univ. (1997).Google Scholar
  145. 145.
    I. I. Pyatetskii-Shapiro, “On the classification of bounded homogeneous domains in the n-dimensional complex space,” Dokl. Akad. Nauk SSSR, 141, 316–319 (1961).MathSciNetGoogle Scholar
  146. 146.
    I. I. Pyatetskii-Shapiro, “Structure of j-algebras,” Izv. Akad. Nauk SSSR, Ser. Mat., 26, 453–484 (1966).Google Scholar
  147. 147.
    H. E. Rauch, “Geodesics and Jacobi equations on homogeneous riemannian manifolds,” in: Proc. United States-Japan Semin. Differ. Geom., Kyoto 1965, Kyoto Univ., Kyoto (1965), pp. 115–127.Google Scholar
  148. 148.
    S. K. Ravindra, “Curvature structures and conformal transformation,” J. Differ. Geom., 4, 425–451 (1969).Google Scholar
  149. 149.
    Yu. G. Reshetnyak, Stability theorems in geometry and analysis, Novosibirsk (1996).Google Scholar
  150. 150.
    A. G. Reznikov, “Weak Blaschke conjecture for HP n,” Dokl. Akad. Nauk SSSR, 283, No. 2, 308–312 (1985).MathSciNetGoogle Scholar
  151. 151.
    E. D. Rodionov, “Homogeneous Riemannian Z-manifolds,” Sib. Mat. Zh., 22, No. 2, 191–197 (1981).zbMATHMathSciNetGoogle Scholar
  152. 152.
    E. D. Rodionov, Structure of Homogeneous Riemannian Z-Manifolds, Dissertation [in Russian], Inst. Math. Sib. Department Russ. Acad. Sci., Novosibirsk (1982).Google Scholar
  153. 153.
    E. D. Rodionov, “Homogeneous Riemannian manifolds of rank 1,” Sib. Mat. Zh., 25, No. 4, 163–166 (1984).zbMATHMathSciNetGoogle Scholar
  154. 154.
    E. D. Rodionov, “Rank of a normal homogeneous space,” Sib. Mat. Zh., 28, No. 5, 154–159 (1987).zbMATHMathSciNetGoogle Scholar
  155. 155.
    E. D. Rodionov, “Geometry of homogeneous Riemannian manifolds,” Dokl. Akad. Nauk SSSR, 306, No. 5, 1049–1051 (1989).MathSciNetGoogle Scholar
  156. 156.
    E. D. Rodionov, “Homogeneous Riemannian almost P-manifods,” Sib. Mat. Zh., 31, No. 5, 102–108 (1990).MathSciNetGoogle Scholar
  157. 157.
    E. D. Rodionov, “Einstein metrics on even-dimensional homogeneous spaces admitting a homogeneous Riemannian metric of positive sectional curvature,” Sib. Mat. Zh., 32, No. 3, 126–131 (1991).MathSciNetGoogle Scholar
  158. 158.
    E. D. Rodionov, “Einstein metrics on a class of 5-dimensional homogeneous spaces,” Commun. Math. Univ. Carolinae, 32, 89–393 (1991).Google Scholar
  159. 159.
    E. D. Rodionov, “On a new family of homogeneous Einstein manifolds,” Arch. Math. (Brno), 28, Nos. 3–4, 199–204 (1992).zbMATHMathSciNetGoogle Scholar
  160. 160.
    E. D. Rodionov, “Homogeneous Einstein metrics on one exceptional Berger space,” Sib. Mat. Zh., 33, No. 1, 208–211 (1992).MathSciNetGoogle Scholar
  161. 161.
    E. D. Rodionov, “Simply connected compact standard homogeneous Einstein manifolds,” Sib. Mat. Zh., 33, No. 4, 104–119 (1992).MathSciNetGoogle Scholar
  162. 162.
    E. D. Rodionov, “Standard homogeneous Einstein manifolds,” Dokl. Ross. Akad. Nauk, 328, No. 2, 147–149 (1993).MathSciNetGoogle Scholar
  163. 163.
    E. D. Rodionov, “Closures of geodesic curves of compact naturally reductive spaces,” in: Proc. Conf Dedicated to the Memory of N. I. Lobachevskii [in Russian], Kazan (1993).Google Scholar
  164. 164.
    E. D. Rodionov, “Simply connected compact five-dimensional homogeneous Einstein manifolds,” Sib. Mat. Zh., 35, No. 1, 163–168 (1994).MathSciNetGoogle Scholar
  165. 165.
    E. D. Rodionov, Homogeneous Riemannian Manifolds with Einstein Metric [in Russian], Dissertation, Inst. Math. Sib. Department Russ. Acad. Sci., Novosibirsk (1994).Google Scholar
  166. 166.
    E. D. Rodionov, “Structure of the standard homogeneous Einstein manifolds with simple isotropy group, I,” Sib. Mat. Zh., 37, No. 1, 175–192 (1996).MathSciNetGoogle Scholar
  167. 167.
    E. D. Rodionov, “Structure of the standard homogeneous Einstein manifolds with simple isotropy group, II,” Sib. Mat. Zh., 37, No. 3, 624–632 (1996).MathSciNetGoogle Scholar
  168. 168.
    E. D. Rodionov and V. V. Slavskii, “Curvature estimations of left-invariant Riemannian metrics on three-dimensional Lie groups,” in: Differ. Geom. Appl. Satellite Conference of ICM in Berlin, Aug. 10–14, 1998, Brno Masaryk University in Brno (Czech Republic), Brno (1999), pp. 111–126.Google Scholar
  169. 169.
    E. D. Rodionov and V. V. Slavskii, “Conformal and rank-one deformations of Riemannian metrics with small areas of zero curvature on a compact manifold,” in: Proc. Conf. “Geometry and Applications” Dedicated to the 70th Birthday of V. A. Toponogov, March 13–16, 2000, Inst. Math. Sib. Department Russ. Acad. Sci., Novosibirsk (2000), pp. 171–182.Google Scholar
  170. 170.
    E. D. Rodionov and V. V. Slavskii, “One-dimensional sectional curvature of Riemannian manifolds,” Dokl. Ross Akad. Nauk, 387, No. 4 (2000).Google Scholar
  171. 171.
    E. D. Rodionov and V. V. Slavskii, “Locally conformally homogeneous Riemannian spaces,” J. ASU, 1, No. 19, 39–42 (2001).Google Scholar
  172. 172.
    E. D. Rodionov and V. V. Slavskii, “Locally conformally homogeneous spaces,” Dokl. Ross. Acad. Nauk, 387, No. 3 (2002).Google Scholar
  173. 173.
    E. D. Rodionov and V. V. Slavskii, “Conformal deformations of the Riemannian metrics and homogeneous Riemannian spaces,” Commun. Math. Univ. Carolinae, 43, No. 2, 271–282 (2002).zbMATHMathSciNetGoogle Scholar
  174. 174.
    E. D. Rodionov, V. V. Slavskii, and L. N. Chibrikova, “Left-invariant Lorenz metrics on Lie groups with zero square of the length of the Schouten-Weil tensor,” Vestn. BGPU, Ser. Estestv. Tochnye Nauki, 4 (2004).Google Scholar
  175. 175.
    V. Yu. Rovenskii and V. A. Toponogov, “Geometric characteristics of the complex projective space,” in: Geometry and Topology of Homogeneous Spaces [in Russian], Barnaul (1988), pp. 98–104.Google Scholar
  176. 176.
    A. Sagle, “On anti-commutative algebras and general Lie triple systems,” Pac. J. Math., 15, 281–291 (1965).zbMATHMathSciNetGoogle Scholar
  177. 177.
    Y. Sakane, “Homogeneous Einstein metrics on flag manifolds,” Lobachevskii J. Math., 4, 71–87 (1999).zbMATHMathSciNetGoogle Scholar
  178. 178.
    H. Sato, “On topological Blaschke conjecture, III,” Lect. Notes Math., 1201, 242–253 (1986).CrossRefGoogle Scholar
  179. 179.
    D. Schueth, “On the ’standard’ condition for noncompact homogeneous Einstein spaces,” Geom. Dedic. (to appear).Google Scholar
  180. 180.
    H. Singh, “On focal locus of submanifolds of naturally reductive compact Riemannian homogeneous spaces,” Proc. Indian Acad. Sci., Math. Sci., 96, No. 2, 131–139 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  181. 181.
    V. V. Slavskii, “Conformally flat metrics of bounded curvature on the n-dimensional sphere,” in: Studies in Geometry “in the Large” and Mathematical Analysis [in Russian], 9, Nauka, Novosibirsk (1987), pp. 183–199.Google Scholar
  182. 182.
    V. V. Slavskii, “Conformally flat metrics and the geometry of the pseudo-Euclidean space,” Sib. Mat. Zh., 35, No. 3, 674–682 (1994).MathSciNetGoogle Scholar
  183. 183.
    M. Stephan, “Conformally flat Lie groups,” Math. Z., 228, 155–175 (1998).zbMATHMathSciNetCrossRefGoogle Scholar
  184. 184.
    V. A. Toponogov, “Extremal theorems for Riemannian spaces of curvature bounded from above,” Sib. Mat. Zh., 15, No. 6, 1348–1371 (1974).zbMATHMathSciNetGoogle Scholar
  185. 185.
    V. A. Topononogov, “Riemannian spaces of diameter equal to π,” Sib. Mat, Zh., 16, No. 1, 124–131 (1975).Google Scholar
  186. 186.
    F. Tricerri, “Locally homogeneous Riemannian manifolds,” Rend. Semin. Mat. Torino, 50, No. 4, 411–426 (1992).zbMATHMathSciNetGoogle Scholar
  187. 187.
    F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds, London Math. Soc. Lect. Note Ser., 83, Cambridge Univ. Press.Google Scholar
  188. 188.
    S. Z. Shefel, “On two classes of k-dimensional surfaces in the n-dimensional Euclidean space,” Sib. Mat. Zh., 10, No. 2, 459–467 (1969).zbMATHMathSciNetGoogle Scholar
  189. 189.
    F. M. Valiev, “Precise estimates for the sectional curvatures of homogeneous Riemannian metrics on Wallach spaces,” Sib. Mat. Zh., 20, 248–262 (1979).zbMATHMathSciNetGoogle Scholar
  190. 190.
    E. B. Vinberg, “Invariant linear connections in a homogeneous manifold,” Tr. Mosk. Mat. Obshch., 9, 191–210 (1960).MathSciNetGoogle Scholar
  191. 191.
    E. B. Vinberg and S. G. Gindikin, “Kähler manifolds admitting a transitive solvable automorphism group,” Mat. Sb., 74, 357–377 (1967).MathSciNetGoogle Scholar
  192. 192.
    D. E. Volper, “Sectional curvatures of a diagonal family of S p(n+1)-invariant metrics on the 4n+3-dimensional spheres,” Sib. Mat. Zh., 35, No. 6, 1089–1100 (1994).MathSciNetCrossRefGoogle Scholar
  193. 193.
    D. E. Volper, “A family of metrics on the 15-dimensional sphere,” Sib. Mat. Zh., 38, No. 2, 263–275 (1997).MathSciNetGoogle Scholar
  194. 194.
    D. E. Volper, “Sectional curvatures of nonstandard metrics on CP 2n+1,” Sib. Mat. Zh., 40, No. 1, 49–56 (1999).MathSciNetGoogle Scholar
  195. 195.
    N. R. Wallach, “Compact homogeneous Riemannian manifolds with strictly positive curvature,” Ann. Math., 96, 277–295 (1972).MathSciNetCrossRefGoogle Scholar
  196. 196.
    M. Wang, “Some examples of homogeneous Einstein manifolds in dimension seven,” Duke Math. J., 49, 23–28 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  197. 197.
    M. Wang, “Einstein metrics from symmetry and bundle constructions,” in: Surv. Differ. Geom.: Essays on Einstein Manifolds, International Press, Cambridge (1999), pp. 287–325.Google Scholar
  198. 198.
    M. Wang and W. Ziller, “On normal homogeneous Einstein manifolds,” Ann. Sci. Ecole Norm. Sup., 18, 563–633 (1985).zbMATHMathSciNetGoogle Scholar
  199. 199.
    M. Wang and W. Ziller, “Existence and non-existence of homogeneous Einstein metrics,” Invent. Math., 84, 177–194 (1986).zbMATHMathSciNetCrossRefGoogle Scholar
  200. 200.
    M. Wang and W. Ziller, “Einstein metrics on principal torus bundles,” J. Differ. Geom., 31, 215–248 (1990).zbMATHMathSciNetGoogle Scholar
  201. 201.
    M. Wang and W. Ziller, “On isotropy irreducible Riemannian manifolds,” Acta Math., 166, 223–261 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  202. 202.
    C. Will, “Rank-one Einstein solvmanifolds of dimension 7,” Differ. Geom. Appl., 19, 307–318 (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  203. 203.
    J. Wolf, “Homogeneity and bounded isometries in manifolds of negative curvature,” Ill. J. Math., 8, 14–18 (1964).zbMATHGoogle Scholar
  204. 204.
    J. Wolf, “The geometry and structure of isotropy irreducible homogeneous spaces,” Acta Math., 120, 59–148 (1968); correction: Acta Math., 152, 141-142 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  205. 205.
    T. H. Wolter, “Einstein metrics on solvable groups,” Math. Z., 206, 457–471 (1991).zbMATHMathSciNetCrossRefGoogle Scholar
  206. 206.
    T. C. Yang, “On the Blaschke conjecture,” Ann. Math. Stud., 102, 159–171 (1982).Google Scholar
  207. 207.
    K. Yano, The Theory of Lie Derivatives and Its Applications, North-Holland Publ., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publ., New York (1957).zbMATHGoogle Scholar
  208. 208.
    W. Ziller, “The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces,” Comment. Math. Helv., 52, 573–590 (1977).zbMATHMathSciNetCrossRefGoogle Scholar
  209. 209.
    W. Ziller, “Homogeneous Einstein metrics on spheres and projective spaces,” Math. Ann., 259, 351–358 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  210. 210.
    W. Ziller, “Weakly symmetric spaces,” in: Progr. Nonlin. Differ. Equations, 20, Birkhauser (1996), pp. 355–368.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Yu. G. Nikonorov
    • 1
  • E. D. Rodionov
    • 2
  • V. V. Slavskii
    • 3
  1. 1.Rubzovsk Industrial InstituteRussia
  2. 2.Barnaul State Pedagogical UniversityRussia
  3. 3.Yugorsk Scientific Research Institute of Informational TechnologiesRussia

Personalised recommendations