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Journal of Mathematical Sciences

, Volume 142, Issue 5, pp 2436–2519 | Cite as

Spaces of Riemannian metrics

  • N. K. Smolentsev
Article

Abstract

In this paper, we consider spaces M of Riemannian metrics on a closed manifold M. In the case where the manifold M is equipped with a symplectic or contact structure, we consider spaces AM of associated metrics. We study geometric and topological properties of these spaces and Riemannian functionals on spaces of metrics.

Keywords

Manifold Riemannian Manifold Scalar Curvature Symplectic Manifold Ahlerian Manifold 
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.

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References

  1. 1.
    R. Abraham, “Transversality in manifolds of mappings,” Bull. Amer. Math. Soc., 69, 470–474 (1963).zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Abraham and J. Marsden. Foundations of Mechanics, Benjamin, New York (1978).zbMATHGoogle Scholar
  3. 3.
    D. V. Alekseevskii, “Compact quaternionic spaces,” Funct. Anal. Appl., 2, 11–20 (1968).MathSciNetGoogle Scholar
  4. 4.
    D. V. Alekseevskii, “Quaternionic Riemannian spaces with a transitive reductive or solvable group of motions,” Funct. Anal. Appl., 4, 68–69 (1970).MathSciNetCrossRefGoogle Scholar
  5. 5.
    D. V. Alekseevskii, “Conjugacy of polar factorizations of Lie groups,” Mat. Sb., 84, 14–26 (1971).MathSciNetGoogle Scholar
  6. 6.
    D. V. Alekseevskii, “Classiffication of quaternionic spaces with a transitive solvable group of motions,” Izv. Akad. Nauk SSSR, Ser. Mat., 25, 93–117 (1975).Google Scholar
  7. 7.
    D. V. Alekseevskii, “Homogeneous Riemannian spaces of negative curvature,” Mat. Sb., 96, 93–117 (1975).MathSciNetGoogle Scholar
  8. 8.
    D. V. Alekseevsky and B. N. Kimelfeld, “Structure of homogeneous Riemannian spaces with zero Ricci curvature,” Funct. Anal. Appl., 9, 27–102 (1975).CrossRefGoogle Scholar
  9. 9.
    D. V. Alekseevsky, I. Doti-Miatello, and C. Ferraris, “Homogeneous Ricci positive 5-manifolds,” Pac. J. Math., 175, 1–12 (1996).Google Scholar
  10. 10.
    M. T. Anderson, “The L 2 structure of moduli spaces of Einstein metrics on 4-manifolds,” Geom. Funct. Anal., 2, No. 1, 29–89 (1992).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    M. T. Anderson, “Degeneration of metrics with bounded curvature and applications to critical metrics of Riemannian functionals,” Proc. Symp. Pure Math., 54, Part 3, 53–79 (1993).Google Scholar
  12. 12.
    V. Apostolov and J. Armstrong, “Symplectic 4-manifolds with HermitianWeil tensor,” Trans. Amer. Math. Soc., 352, 4501–4513 (2000).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    V. Apostolov, J. Armstrong, and T. Draghici, “Local rigidity of certain classes of almost Kähler 4-manifolds,” Ann. Global Anal. Geom., 21, 151–176 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    V. Apostolov and T. Draghici, “Almost Kähler 4-manifolds with J-invariant Ricci tensor and special Weil tensor,” Quart. J. Math., 51, 275–294 (2000).zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Araujo, “Critical points of the total scalar curvature plus total mean curvature functional,” Indiana Univ. Math. J., 52, No. 1, 85–107 (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Arvanitoyergos, “SO(n)-invariant Einstein metrics on Stiefel manifolds,” in: Proc. Conf. Diff. Geom. Appl., Brno, 1995, pp. 1–5.Google Scholar
  17. 17.
    A. Arvanitoyergos, “Einstein equations for invariant metric on generalized flag manifolds and inner automorphism,” Balkan J. Geom. Appl., 1, 17–22 (1996).Google Scholar
  18. 18.
    V. I. Averbukh and O. G. Smolyanov, “Theory of differentiation in topological vector spaces,” Usp. Mat. Nauk, 22, No. 6, 201–260 (1967).Google Scholar
  19. 19.
    V. I. Averbukh and O. G. Smolyanov, “Definitions of the derivative in topological vector spaces,” Usp. Mat. Nauk, 23, No. 4, 67–116 (1968).Google Scholar
  20. 20.
    A. Avez, “Applications de la formule de Gauss-Bonnet-Chern aux varietes à quatre dimensions,” C. R. Acad. Sci. Paris, 256, 5488–5490 (1963).zbMATHMathSciNetGoogle Scholar
  21. 21.
    L. Bérard Bergery, M. Berger, and C. Houzel, eds., Géométrie Riemannienne en Dimension 4. Seminaire Arthur Besse 1978/79, CEDIC/Fernand Nathan, Paris (1981).zbMATHGoogle Scholar
  22. 22.
    M. Berger, “Sur la spectre d’une variete Riemannienne,” C. R. Acad. Sci. Paris, 263, 13–16 (1963).Google Scholar
  23. 23.
    M. Berger, “Quelques formules de variation pour une structure Riemannienne,” Ann. Sci. Ecole Norm. Super., IV. Ser., 3, No. 3, 285–294 (1970).zbMATHGoogle Scholar
  24. 24.
    M. Berger, P. Gauduchon, and E. Mazet, “Le spectre d’une variete Riemannienne,” Lect. Notes Math., 194 (1971).Google Scholar
  25. 25.
    M. Berger and D. Ebin, “Some decompositions of the space of symmetric tensors on a Riemannian manifold,” J. Differ. Geom., 3, No. 3, 379–392 (1969).zbMATHMathSciNetGoogle Scholar
  26. 26.
    A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin (1987).zbMATHGoogle Scholar
  27. 27.
    D. E. Blair, “Contact manifolds in Riemannian geometry,” Lect. Notes Math., 509 (1976).Google Scholar
  28. 28.
    D. E. Blair, “On the space of Riemannian metrics on surfaces and contact manifolds,” Lect. Notes Math., 792, 203–219 (1980).MathSciNetCrossRefGoogle Scholar
  29. 29.
    D. E. Blair, “On the set of metrics associated to a symplectic or contact forms,” Bull. Inst. Math. Sinica, 11, 297–308 (1983).zbMATHMathSciNetGoogle Scholar
  30. 30.
    D. E. Blair, “Critical associated metrics on contact manifolds,” J. Austr. Math. Soc., Ser. A, 37, 82–88 (1984).zbMATHMathSciNetGoogle Scholar
  31. 31.
    D. E. Blair, “Critical associated metrics on contact manifolds, III,” J. Austr. Math. Soc., Ser. A, 50, 189–196 (1991).zbMATHMathSciNetGoogle Scholar
  32. 32.
    D. E. Blair, “The ‘total scalar curvature’ as a symplectic invariant,” in: Proc. 3 Congr. Geom., Thessaloniki (1991), pp. 79–83.Google Scholar
  33. 33.
    D. E. Blair, “Spaces of metrics and curvature functionals,” in: Handbook of Differential Geometry, Vol. 1, North-Holland, Amsterdam (2000), pp. 153–158.CrossRefGoogle Scholar
  34. 34.
    D. E. Blair and S. Ianus, “Critical associated metrics on symplectic manifolds,” Contemp. Math., 51, 23–29 (1986).MathSciNetGoogle Scholar
  35. 35.
    D. E. Blair and J. Ledger, “Critical associated metrics on contact manifolds, II,” J. Austr. Math. Soc., Ser. A, 43, 404–410 (1986).MathSciNetGoogle Scholar
  36. 36.
    D. E. Blair and D. Perrone, “A variational characterization of contact metric manifolds with vanishing torsion,” Can. Math. Bull., 35, 455–462 (1992).zbMATHMathSciNetGoogle Scholar
  37. 37.
    D. E. Blair and D. Perrone, “Second variation of the ‘total scalar curvature’ on contact manifolds,” Can. Math. Bull., 38, 16–22 (1995).zbMATHMathSciNetGoogle Scholar
  38. 38.
    D. D. Bleecker, “Critical Riemannian manifolds,” J. Differ. Geom., 14, 599–608 (1979).zbMATHMathSciNetGoogle Scholar
  39. 39.
    C. Böhm and M. Kerr, Low-dimensional homogeneous Einstein manifolds, preprint (2002).Google Scholar
  40. 40.
    C. Böhm, M. Wang, and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, preprint (2002).Google Scholar
  41. 41.
    C. Böhm. Homogeneous Einstein metrics and simplicial complexes, preprint (2003).Google Scholar
  42. 42.
    S. Bochner, “Vectors fields and Ricci curvature,” Bull. Ann. Math. Soc., 52, 776–797 (1946).zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    J.-P. Bourguignon, “Une stratification de l’espace des structures Riemanniennes,” Commun. Math., 30, 1–41 (1975).zbMATHMathSciNetGoogle Scholar
  44. 44.
    J.-P. Bourguignon, D. G. Ebin, and J. E. Marsden, “Sur le noyau des pseudo-differentiels à symbole surjectif et non injectif,” C. R. Acad. Sci. Paris, Ser. A, 282, 867–870 (1976).zbMATHMathSciNetGoogle Scholar
  45. 45.
    J.-P. Bourguignon and J. P. Ezin, “Scalar curvature functions in a conformal class of metrics and conformal transformations,” Trans. Amer. Math. Soc., 310, No. 2, 867–870 (1987).MathSciNetGoogle Scholar
  46. 46.
    D. Burns and P. de Bartolomeis, “Stability of vector bundles and extremal metrics,” Invent. Math., 92, 403–407 (1988).zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    E. Calabi, “The space of Kähler metrics,” in: Proc. Int. Congr. Math., Vol. 2, Amsterdam (1954), pp. 206–207.Google Scholar
  48. 48.
    E. Calabi, “Extremal Kähler metrics,” Ann. Math. Stud., 102, 259–290 (1982).MathSciNetGoogle Scholar
  49. 49.
    E. Calabi, “Extremal Kähler metrics, II,” in: Differential Geometry and Complex Analysis, Springer, Berlin (1985), pp. 95–114.Google Scholar
  50. 50.
    S. S. Chern and J. Simons, “Characteristic forms and geometric invariants,” Ann. Math., 99, 48–69 (1974).MathSciNetCrossRefGoogle Scholar
  51. 51.
    S. S. Chern and R. S. Hamilton, “On Riemannian metrics adopted to three-dimensional contact manifolds,” Lect. Notes Math., 1111, 279–308 (1985).MathSciNetCrossRefGoogle Scholar
  52. 52.
    J. Davidov and O. Muskarov, “Twistor spaces with Hermitian Ricci tensor,” Proc. Amer. Math. Soc., 109, 1115–1120 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    N. A. Daurtseva and N. K. Smolentsev, On the space of almost complex structures, Preprint mathDG/0202139, http://xxx.lanl.gov.Google Scholar
  54. 54.
    N. A. Daurtseva, U(n+1) × U(p+1)-Invariant Hermitian metrics with Hermitian tensor Ricci on the manifold S 2n+1 × S 2p+1, Preprint mathDG/0310124, http://xxx.lanl.gov.Google Scholar
  55. 55.
    P. Delanoe, “On Bianchi identities,” Rend. Circ. Mat. Palermo, 51, Ser. 2, 237–248 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    B. S. DeWitt, “Quantum theory of gravity. I. The canonical theory,” Phys. Rev., 160, 1113–1148 (1967).zbMATHCrossRefGoogle Scholar
  57. 57.
    T. Draghici, “On the almost Kähler 4-manifolds with Hermitian Ricci tensor,” Houston J. Math., 20, No. 2, 293–298 (1994).zbMATHMathSciNetGoogle Scholar
  58. 58.
    T. Draghici, “On some 4-dimensional almost Kähler manifolds,” Kodai Math. J., 18, 156–168 (1995).zbMATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    T. Draghici, “Almost Kähler 4-manifolds with J-invariant Ricci tensor,” Houston J. Math., 25, 133–145 (1999).zbMATHMathSciNetGoogle Scholar
  60. 60.
    C. J. Earle and J. Eells, “A fibre bundle description of Teichmüller theory,” J. Differ. Geom., 3, 19–43 (1969).zbMATHMathSciNetGoogle Scholar
  61. 61.
    J. Eells, “On the geometry of function spaces,” in: Symposium de Topologia Algebra, Mexico (1958), pp. 303–307.Google Scholar
  62. 62.
    J. Eells, “On submanifolds of certain function spaces,” Proc. Nat. Acad. Sci., 45, No. 10, 1520–1522 (1959).zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    D. Ebin, “The manifold of Riemannian metrics,” Proc. Symp. Pure Math., 15, 11–40 (1970).MathSciNetGoogle Scholar
  64. 64.
    D. Ebin and J. Marsden, “Groups of diffeomorphisms and the motion of an incompressible fluid,” Ann. Math., 92, No. 1, 102–163 (1970).MathSciNetCrossRefGoogle Scholar
  65. 65.
    J. Eichorn, “The Banach manifold structure of the space of metrics on noncompact manifolds,” Differ. Geom. Appl., 1, 89–108 (1991).CrossRefGoogle Scholar
  66. 66.
    J. Eichorn, “Spaces of Riemannian metrics on open manifolds,” Results Math., 27, 256–283 (1995).MathSciNetGoogle Scholar
  67. 67.
    J. Eichorn, “Diffeomorphism groups on noncompact manifolds,” Zap. Nauch. Semin. POMI, 234, 41–64 (1996).Google Scholar
  68. 68.
    J. Eichorn, “Poincare’s theorem and Teichmüller theory for open surfaces,” Asian J. Math., 2, No. 2, 355–403 (1998).MathSciNetGoogle Scholar
  69. 69.
    J. Eichorn, “A classiffication approach for open manifolds,” Zap. Nauch. Semin. POMI, 267, 9–45 (2000).Google Scholar
  70. 70.
    H. I. Eliasson, “On the geometry of manifold of maps,” J. Differ. Geom., 1, 169–194 (1967).zbMATHMathSciNetGoogle Scholar
  71. 71.
    H. I. Eliasson, “On variations of metrics,” Math. Scand., 29, 317–372 (1971).MathSciNetGoogle Scholar
  72. 72.
    J. F. Escobar, “The Yamabe problem on manifolds with boundary,” J. Differ. Geom., 35, 21–84 (1992).zbMATHMathSciNetGoogle Scholar
  73. 73.
    J. F. Escobar, “Conformal deformation of Riemannian metric to a scalar flat metric with constant mean curvature on the boundary,” Ann. Math., 136, No. 2, 1–50 (1992).MathSciNetGoogle Scholar
  74. 74.
    J. F. Escobar, “Conformal deformation of Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary,” Indiana Univ. Math. J., 45, 917–943 (1996).zbMATHMathSciNetCrossRefGoogle Scholar
  75. 75.
    A. Fathi and L. Flaminio, “Infinitesimal conjugacies and Weil-Petersson metric,” Ann. Inst. Fourier, 43, No. 1, 279–299 (1993).zbMATHMathSciNetGoogle Scholar
  76. 76.
    A. Fischer, “The theory of superspaces,” in: Relativity, Plenum Press, New York (1970), pp. 303–357.Google Scholar
  77. 77.
    A. Fischer, “Unfolding the singularities in superspaces,” Gen. Relativity Gravitation, 15, 1191–1198 ().Google Scholar
  78. 78.
    A. Fischer, “Resolving the singularities in the space of Riemannian geometries,” J. Math. Phys., 27, No. 3, 718–738 (1986).zbMATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    A. Fischer and J. Marsden, “Linearization stabillity of nonlinear partial differential equations,” Proc. Symp. Pure Math., 27, Part 2, 219–263 (1975).MathSciNetGoogle Scholar
  80. 80.
    A. Fischer and J. Marsden, “Deformations of the scalar curvature,” Duke Math. J., 42, 519–547 (1975).zbMATHMathSciNetCrossRefGoogle Scholar
  81. 81.
    A. Fischer and J. Marsden, “The manifold of conformally equivalent metrics,” Can. J. Math., 29, No. 1, 193–209 (1977).zbMATHMathSciNetGoogle Scholar
  82. 82.
    A. Fischer and A. Tromba, “On a purely ‘Riemannian’ proof of the structure and dimension of the unramified moduli space of a compact Riemannian surface,” Math. Ann., 267, 311–345 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  83. 83.
    A. Fischer and A. Tromba, “Almost complex principal fiber bundles and the complex structure on Teichmüller space,” J. Reine Angew. Math., 352, 151–160 (1984).zbMATHMathSciNetGoogle Scholar
  84. 84.
    A. Fischer and A. Tromba, “On the Weil-Petersson metric on Teichmüller space,” Trans. Amer. Math. Soc., 284, No. 1, 329–335 (1984).MathSciNetCrossRefGoogle Scholar
  85. 85.
    A. Fischer and A. Tromba, “A new proof that Teichmüller space is a cell,” Trans. Amer. Math. Soc., 303, No. 1, 257–262 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  86. 86.
    D. S. Freed and D. Groisser, “The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group,” Michigan Math. J., 36, No. 3, 323–344 (1989).zbMATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    A. Fujiki, “Coarse moduli spaces for polarized Kähler manifolds,” Publ. RIMS, Kyoto Univ., 20, 977–1005 (1984).zbMATHMathSciNetGoogle Scholar
  88. 88.
    A. Fujiki, “Remarks on extremal Kähler metrics on ruled manifolds,” Nagoya Math. J., 126, 89–101 (1992).zbMATHMathSciNetGoogle Scholar
  89. 89.
    A. Fujiki and G. Schumacher, “The moduli space of Kähler structures on a real compact symplectic manifold,” Publ. RIMS, Kyoto Univ., 24, No. 1, 141–168 (1988).zbMATHMathSciNetGoogle Scholar
  90. 90.
    A. Fujiki and G. Schumacher, “The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics,” Publ. RIMS, Kyoto Univ., 26, 101–183 (1990).zbMATHMathSciNetGoogle Scholar
  91. 91.
    A. Futaki, “An obstruction to the existence of Einstein Kähler metrics,” Invent. Math., 73, No. 3, 438–443 (1983).MathSciNetCrossRefGoogle Scholar
  92. 92.
    A. Futaki, “On compact Kähler manifolds of constant scalar curvature,” Proc. Japan Acad. Ser. A, 59, 401–402 (1983).zbMATHMathSciNetGoogle Scholar
  93. 93.
    A. Futaki and T. Mabuchi, “Bilinear forms and extremal Kähler vector fields associated with Kähler classes,” Math. Ann., 301, 199–210 (1995).zbMATHMathSciNetCrossRefGoogle Scholar
  94. 94.
    O. Gil-Medrano and P. Michor, “The Riemannian manifold of all Riemannian metrics,” Quart. J. Math. Oxford, 42, No. 2, 183–202 (1991).zbMATHMathSciNetCrossRefGoogle Scholar
  95. 95.
    P. Gauduchon, “Variation des courbures scalaires en geometrie hermitienne,” C. R. Acad. Sci. Paris, 290, 327–330 (1980).zbMATHMathSciNetGoogle Scholar
  96. 96.
    P. Gauduchon, “La 1-forme de torsion d’une variete hermitienne compacte,” Math. Ann., 267, 495–518 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  97. 97.
    P. Gilkey, “The spectral geometry of a Riemannian manifold,” J. Differ. Geom., 10, 601–618 (1975).zbMATHMathSciNetGoogle Scholar
  98. 98.
    S. I. Goldberg, “Integrability of almost Kähler manifolds,” Proc. Amer. Math. Soc., 21, 96–100 (1969).zbMATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    C. S. Gordon and M. Kerr, “New homogeneous metrics of negative Ricci curvature,” Ann. Global Anal. Geom., 19, 1–27 (2001).MathSciNetCrossRefGoogle Scholar
  100. 100.
    C. S. Gordon and Z. I. Szabo, “Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric,” Duke Math. J., 113, No. 2, 355–383 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  101. 101.
    H. Grauert and R. Remmert, “Über kompakte homogene komplexe Mannigfaltigkeiten,” Arch. Math., 13, 498–507 (1962).zbMATHMathSciNetCrossRefGoogle Scholar
  102. 102.
    J. Gravesen, “Complex structures in the Nash-Moser category,” Ann. Global Anal. Geom., 7, No. 2, 155–161 (1989).zbMATHMathSciNetCrossRefGoogle Scholar
  103. 103.
    D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Großen, Lect. Notes Math., 55, Springer-Verlag, Berlin-Heidelberg-New York (1975).zbMATHGoogle Scholar
  104. 104.
    M. Gromov, “Pseudoholomorphic curves in symplectic manifolds,” Invent. Math., 82, No. 2, 307–347 (1985).zbMATHCrossRefMathSciNetGoogle Scholar
  105. 105.
    M. Gromov, “Soft and hard symplectic geometry,” in: Proc. Int. Congr. Math., (1987), pp. 81–98.Google Scholar
  106. 106.
    V. Guillemin and S. Sternberg, “Remarks on a paper of Hermann,” Trans. Amer. Math. Soc., 130, 110–116 (1968).zbMATHMathSciNetCrossRefGoogle Scholar
  107. 107.
    R. S. Hamilton, “The inverse function theorem of Nash and Moser,” Bull. Amer. Math. Soc., 7, No. 1, 65–222 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  108. 108.
    J. Heber, “Noncompact homogeneous Einstein spases,” Invent. Math., 133, 279–352 (1998).zbMATHMathSciNetCrossRefGoogle Scholar
  109. 109.
    S. Helgason, Differential Geometry and Symmetric Spaces, Pure Appl. Math., 12, Academic Press, New York-London (1962).Google Scholar
  110. 110.
    R. Hermann, “The formal linearization of a semi-simple Lie algebra of vector fields about a singular point,” Trans. Amer. Math. Soc., 130, 105–109 (1968).zbMATHMathSciNetCrossRefGoogle Scholar
  111. 111.
    D. Hilbert, “Die grundlagen der Physik,” Nachr. Ges. Wiss. Gottingen, 395–407 (1915).Google Scholar
  112. 112.
    A. D. Hwang, “Extremal Kähler metrics and the Calabi energy,” Proc. Japan Acad. Ser. A, Math. Sci., 71, 128–129 (1995).zbMATHMathSciNetGoogle Scholar
  113. 113.
    A. D. Hwang, “On the Calabi energy of extremal Kähler metrics,” Int. J. Math., 6, No. 6, 825–830 (1995).zbMATHCrossRefGoogle Scholar
  114. 114.
    A. D. Hwang and S. G. Simanca, “Distinguished Kähler metrics on Hirzebruch surfaces,” Trans. Amer. Math. Soc., 347, No. 1, 1013–1021 (1995).zbMATHMathSciNetCrossRefGoogle Scholar
  115. 115.
    S. Ishihara, “Homogeneous Riemannian spaces of four dimensions,” J.Math. Soc. Japan, 7, 345–370 (1955).zbMATHMathSciNetGoogle Scholar
  116. 116.
    M. Itoh, “Conformal geometry of Ricci flat 4-manifolds,” Kodai Math. J., 17, 179–200 (1994).zbMATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    G. R. Jensen, “Homogeneous Einstein spaces of dimension 4,” J. Differ. Geom., 3, 309–349 (1969).zbMATHGoogle Scholar
  118. 118.
    G. R. Jensen, “The scalar curvature of left invariant Riemannian metrics,” Indiana Univ. Math. J., 20, 1125–1143 (1971).MathSciNetCrossRefGoogle Scholar
  119. 119.
    G. R. Jensen, “Einstein metrics on principal fibre bundles,” J. Differ. Geom., 8, 599–614 (1973).zbMATHGoogle Scholar
  120. 120.
    J. L. Kazdan, “Another proof of Bianchi’s identity in Riemannian geometry,” Proc. Amer. Math. Soc., 81, No. 2, 341–342 (1981).zbMATHMathSciNetCrossRefGoogle Scholar
  121. 121.
    J. L. Kazdan and F. W. Warner, “Scalar curvature and conformal deformation of Riemannian structure,” J. Differ. Geom., 10, No. 1, 113–134 (1975).zbMATHMathSciNetGoogle Scholar
  122. 122.
    J. L. Kazdan and F. W. Warner, “Existence and conformal deformation of of metrics with prescribed Gaussian and scalar curvatures,” Ann. Math., 101, 317–331 (1975).MathSciNetCrossRefGoogle Scholar
  123. 123.
    J. L. Kazdan and F. W. Warner, “Prescribing curvatures,” Proc. Symp. Pure Math., 27, Part 2, 309–319 (1975).MathSciNetGoogle Scholar
  124. 124.
    J. L. Kazdan and F. W. Warner, “A direct approach to the determination of Gaussian and scalar curvature functions,” Invent. Math., 28, No. 3, 227–230 (1975).zbMATHMathSciNetCrossRefGoogle Scholar
  125. 125.
    M. Kerr, “Some new homogeneous Einstein metrics on symmetric spaces,” Trans. Amer. Math. Soc., 348, 153–171 (1996).zbMATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    M. Kerr, “New examples of homogeneous Einstein metrics,” Michigan J. Math., 45, 115–134 (1998).zbMATHMathSciNetCrossRefGoogle Scholar
  127. 127.
    M. Kerr, A deformation of quaternionic hyperbolic space, Preprint (2002).Google Scholar
  128. 128.
    B. H. Kim, “Warped products with critical Riemannian metric,” Proc. Japan Acad. Ser. A, Math. Sci., 71, 117–118 (1995).zbMATHMathSciNetGoogle Scholar
  129. 129.
    J. Kim and C. Sung, “Deformations of almost Kähler metrics with constant scalar curvature on compact Kähler manifolds,” Ann. Global Anal. Geom., 22, 49–73 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  130. 130.
    O. Kobayashi, “A differential equation arising from scalar curvature function,” J. Math. Soc. Japan, 34, No. 4, 665–675 (1982).zbMATHMathSciNetGoogle Scholar
  131. 131.
    O. Kobayashi, “On a conformally invariant functional of the space of Riemannian metrics,” J. Math. Soc. Japan, 37, No. 3, 373–389 (1985).zbMATHMathSciNetGoogle Scholar
  132. 132.
    Sh. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin-Heidelberg-New York (1972).zbMATHGoogle Scholar
  133. 133.
    Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1, Interscience Publishers, New York-London (1963); Vol. 2 (1969).zbMATHGoogle Scholar
  134. 134.
    T. Koda, “Critical almost Hermitian structures,” Indian J. Pure Appl. Math. Soc., 26, 679–690 (1995).zbMATHMathSciNetGoogle Scholar
  135. 135.
    N. Koiso, “Non-deformability of Einstein metrics,” Osaka J. Math., 15, 419–433 (1978).zbMATHMathSciNetGoogle Scholar
  136. 136.
    N. Koiso, “Einstein metrics and complex structures,” Invent. Math., 73, No. 1, 71–106 (1983).zbMATHMathSciNetCrossRefGoogle Scholar
  137. 137.
    N. Koiso, “A decomposition of the space M of Riemannian metrics on a manifold,” Osaka J. Math., 16, 423–429 (1979).zbMATHMathSciNetGoogle Scholar
  138. 138.
    N. H. Kuiper, “On compact conformally Euclidean spaces of dimension >2,” Ann. Math., 52, 478–490 (1950).MathSciNetCrossRefGoogle Scholar
  139. 139.
    J. Lafontaine, “Sur la geometrie d’une generalisation de l’equation d’Obata,” J. Math. Pures Appl., 62, 63–72 (1983).zbMATHMathSciNetGoogle Scholar
  140. 140.
    F. Lamontagne, “Une remarque sur la norme L 2 du tenseur de courbure,” C. R. Acad. Sci. Paris, 319, 237–240 (1994).zbMATHMathSciNetGoogle Scholar
  141. 141.
    S. Lang, Differential Manifolds, Addison-Wesley, Reading, Massachusetts (1972).zbMATHGoogle Scholar
  142. 142.
    C. LeBrun, Explicit self-dual metrics on ℂP 2#…ℂP 2, Preprint, Princeton (1989).Google Scholar
  143. 143.
    C. LeBrun and S. Simanca, “On the Kähler classes of extremal metrics,” in: Geom. Global Anal., Sendai (1993), pp. 255–271.Google Scholar
  144. 144.
    C. LeBrun and S. Simanca, “Extremal Kähler metrics and complex deformation theory,” Geom. Funct. Anal., 4, 298–336 (1994).zbMATHMathSciNetCrossRefGoogle Scholar
  145. 145.
    J. Leslie, “On a differential structure for the group of diffeomorphisms,” Topology, 6, 263–271 (1967).zbMATHMathSciNetCrossRefGoogle Scholar
  146. 146.
    M. Levin, “A remark on extremal Kähler metrics,” J. Differ. Geom., 21, 73–77 (1985).Google Scholar
  147. 147.
    A. Lihnerowicz, “Propogateurs et commutateurs en relative generale,” Publ. Math. Inst. des Hautes Etudes Sci., 10, 293–344 (1961).Google Scholar
  148. 148.
    A. Lihnerowicz, “Spineurs harmoniques,” C. R. Acad. Sci. Paris, 257, 7–9 (1963).MathSciNetGoogle Scholar
  149. 149.
    D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys., 12, 498–501 (1971).zbMATHMathSciNetCrossRefGoogle Scholar
  150. 150.
    L. Maxim-Raileanu, “The manifold of Riemannian metrics on a compact manifold with boundary,” An. St. Univ. “Al. I. Cusa” Iasi. Ser. 1a, 34, No. 2, 67–72 (1988).zbMATHGoogle Scholar
  151. 151.
    L. Maxim-Raileanu, “Critical mapping of Riemannian manifolds and the slice theorem for an action of the diffeomorphism group,” An. St. Univ. “Al. I. Cusa” Iasi. Ser. 1a, 34, No. 2, 149–152 (1988).zbMATHMathSciNetGoogle Scholar
  152. 152.
    H. P. McKean and I. M. Singer, “Curvature and the eigenvalues of the Laplacian,” J. Differ. Geom., 1, 43–69 (1967).zbMATHMathSciNetGoogle Scholar
  153. 153.
    P. Michor, “Manifolds of smooth maps,” Cahiers Topol. Geom. Differ., 19, No. 1, 47–78 (1978).MathSciNetzbMATHGoogle Scholar
  154. 154.
    P. Michor, “Manifolds of smooth maps, II. The Lie group of diffeomorphisms of a non-compact smooth manifolds,” Cahiers Topol. Geom. Differ., 21, No. 1, 63–86 (1980).MathSciNetzbMATHGoogle Scholar
  155. 155.
    J. W. Milnor, “Eigenvalues of the Laplace operator on certain manifolds,” Proc. Nat. Acad. Sci. USA, 51, 542 (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  156. 156.
    J. W. Milnor, “Remarks on infinite-dimensional Lie groups,” in: Relativity, Groups, and Topology, II, Les Houches Session XL, 1983 (B. S. de Witt and R. Stora, Eds.), North-Holland, Amsterdam (1984).Google Scholar
  157. 157.
    S. Minakshisundaram and A. Pleijel, “Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds,” Can. J. Math., 1, 242–256 (1949).zbMATHMathSciNetGoogle Scholar
  158. 158.
    J. Moser, “On the volume elements on a manifold,” Trans. Amer. Math. Soc., 120, 286–294 (1965).zbMATHMathSciNetCrossRefGoogle Scholar
  159. 159.
    O. Muskarov, “On Hermitian surfaces with J-invariant Ricci tensor,” J. Geom., 72, 151–156 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  160. 160.
    Y. Muto, “On Einstein metrics,” J. Differ. Geom., 9, No. 4, 521–530 (1974).zbMATHMathSciNetGoogle Scholar
  161. 161.
    Y. Muto, “Curvature and critical Riemannian metric,” J. Math. Soc. Japan, 26, No. 4, 686–897 (1974).zbMATHMathSciNetGoogle Scholar
  162. 162.
    Y. Muto, “Critical Riemannian metrics,” Tensor N. S., 29, 125–133 (1975).zbMATHMathSciNetGoogle Scholar
  163. 163.
    Y. Muto, “Critical Riemannian metrics on product manifolds,” Kodai Math. Sem. Rep., 26, 409–423 (1975).zbMATHMathSciNetCrossRefGoogle Scholar
  164. 164.
    Y. Muto, “Curvature and critical Riemannian metric,” Proc. Symp. Pure Math., 27, Part 1, 97–100 (1975).MathSciNetGoogle Scholar
  165. 165.
    Y. Muto, “Riemannian submersions and critical Riemannian metric,” J. Math. Soc. Japan, 29, 493–511 (1977).zbMATHMathSciNetGoogle Scholar
  166. 166.
    S. B. Myers, “Riemannian manifolds with positive mean curvature,” Duke Math. J., 8, 401–404 (1941).zbMATHMathSciNetCrossRefGoogle Scholar
  167. 167.
    M. Neuwirther, “Submanifold geometry and Hessians on the pseudoriemannian manifold of metrics,” Acta Math. Univ. Comenianae, 62, No. 1, 51–85 (1993).zbMATHMathSciNetGoogle Scholar
  168. 168.
    Yu. G. Nikonorov, “Functional of scalar curvature and homogeneous Einsteinian metrics on Lie groups,” Sib. Mat. Zh., 39, No. 3, 583–589 (1998).zbMATHMathSciNetGoogle Scholar
  169. 169.
    Yu. G. Nikonorov, “New series of Einstein homogeneous metrics,” Differ. Geom. Appl., 12, 25–34 (2000).zbMATHMathSciNetCrossRefGoogle Scholar
  170. 170.
    Yu. G. Nikonorov, “On one class of homogeneous compact Einstein manifolds,” Sib. Mat. Zh., 41, No. 1, 200–205 (2000).zbMATHMathSciNetGoogle Scholar
  171. 171.
    Yu. G. Nikonorov, “Algebraic structure of standard homogeneous Einstein manifolds,” Mat. Tr., 3, No. 1, 119–143 (2000).zbMATHMathSciNetGoogle Scholar
  172. 172.
    Yu. G. Nikonorov, “On the Ricci curvature of homogeneous metrics on noncompact homogeneous space,” Sib. Mat. Zh., 41, No. 2, 421–429 (2000).zbMATHMathSciNetGoogle Scholar
  173. 173.
    Yu. G. Nikonorov, “Compact seven-dimensional homogeneous Einstein manifolds,” Dokl. Ross. Akad. Nauk, 372, No. 5, 589–592 (2000).MathSciNetGoogle Scholar
  174. 174.
    Yu. G. Nikonorov, “Classification of invariant Einstein metrics on the Aloff-Wallach spaces,” Sib. Adv. Math., 13, No. 4, 70–89 (2003).zbMATHMathSciNetGoogle Scholar
  175. 175.
    Yu. G. Nikonorov, “Invariant Einstein metrics on the Ledger-Obata spaces,” St. Petersburg Math. J., 14, No. 3, 487–497 (2003).MathSciNetGoogle Scholar
  176. 176.
    Yu. G. Nikonorov and E. D. Rodionov, “Standard homogeneous Einstein manifolds and Diophantine equations,” Arch. Math., 32, 123–136 (1996).zbMATHMathSciNetGoogle Scholar
  177. 177.
    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
  178. 178.
    Yu. G. Nikonorov and E. D. Rodionov, “Compact homogeneous Einstein 6-manifolds,” Differ. Geom. Appl., 19. 369–378 (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  179. 179.
    P. Nurowski and M. Przanowski, “A four-dimensional example of Ricci flat metric admitting almost Kähler non-Kähler structure,” Class. Quantum Gravit., 16, L9–L13 (1999).zbMATHMathSciNetCrossRefGoogle Scholar
  180. 180.
    M. Obata, “Certain conditions for a Riemannian manifold to be isometric to a sphere,” J. Math. Soc. Japan, 14, 333–340 (1962).zbMATHMathSciNetCrossRefGoogle Scholar
  181. 181.
    H. Omori, “On the group of diffeomorphisms on a compact manifold,” Proc. Symp. Pure Math., 15, 167–183 (1970).MathSciNetGoogle Scholar
  182. 182.
    H. Omori, Infinite-Dimensional Lie Transformations Groups, Lect. Notes Math., 427 (1974).Google Scholar
  183. 183.
    R. S. Palais, “On the differentiability of isometries,” Proc. Amer. Math. Soc., 8, 805–807 (1957).zbMATHMathSciNetCrossRefGoogle Scholar
  184. 184.
    R. S. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton Univ. Press, Princeton, New Jersey (1965).zbMATHGoogle Scholar
  185. 185.
    R. S. Palais, “The principle of symmetric criticality,” Commun. Math. Phys., 69, 19–30 (1979).zbMATHMathSciNetCrossRefGoogle Scholar
  186. 186.
    R. S. Palais, “Applications of the symmetric criticality principle to mathematical physics and differential geometry,” in: Proc. Symp. Differ. Geom. Differ. Equations, Shanghai-Hefei 1981 (1985), pp. 247–302.Google Scholar
  187. 187.
    R. S. Palais, Foundations of Global Nonlinear Analysis, Benjamin, New York (1968).Google Scholar
  188. 188.
    E. M. Patterson, “A class of critical Riemannian metrics,” J. London Math. Soc., 2, 349–358 (1981).MathSciNetCrossRefGoogle Scholar
  189. 189.
    H. Pedersen and Y. S. Poon, “Hamiltonian construction of Kähler-Einstein metrics and metrics of constant scalar curvature,” Commun. Math. Phys., 136, No. 2, 309–326 (1991).zbMATHMathSciNetCrossRefGoogle Scholar
  190. 190.
    O. Pekonen, “On the De Witt metric,” J. Geom. Phys., 4, No. 4, 493–502 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  191. 191.
    O. Pekonen, “On the variational characterization of conformally flat 3-manifolds,” J. Geom. Phys., 7, No. 1, 109–117 (1987).MathSciNetCrossRefGoogle Scholar
  192. 192.
    O. Pekonen, “A short proof of a theorem of Ahlfors,” Lect. Notes in Math., 1351, 273–278 (1988).MathSciNetCrossRefGoogle Scholar
  193. 193.
    O. Pekonen, “Slice theorem for the action of the diffeomorphism groups on the space of almost complex structures on a compact manifold,” Bull. Polish Acad. Sci. Math., 37, Nos. 7–12, 545–548 (1989).zbMATHMathSciNetGoogle Scholar
  194. 194.
    D. Perrone, “Torsion and critical metrics on contact three-manifolds,” Kodai Math. J., 13, 88–100 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  195. 195.
    D. Perrone, “Torsion tensor and critical metrics on contact 2n+1-manifolds,” Monatsh. Math., 114, 245–259 (1992).zbMATHMathSciNetCrossRefGoogle Scholar
  196. 196.
    A. Pressley and G. Segal, Loop Groups, Oxford Math. Monogr., Clarendon Press, Oxford (1988).zbMATHGoogle Scholar
  197. 197.
    T. Ratiu and R. Shmid, “The differentiable structure of three remarkable diffeomorphism groups,” Math. Z., 177, 81–100 (1981).zbMATHMathSciNetCrossRefGoogle Scholar
  198. 198.
    E. D. Rodionov, “Standard homogeneous Einstein manifolds,” Dokl. Ross. Akad. Nauk, 328, No. 2, 147–149 (1993).MathSciNetGoogle Scholar
  199. 199.
    R. Schoen, “Conformal deformations of Riemannian metrics to constant scalar curvature,” J. Differ. Geom., 20, 479–495 (1984).zbMATHMathSciNetGoogle Scholar
  200. 200.
    D. Schueth, “On the ’standard’ condition for noncompact homogeneous Einstein spaces,” Geom. Dedicata, 105, 77–83 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  201. 201.
    G. Schumacher, “Construction of the coarse moduli space of compact polarized Kähler manifolds with c 1 = 0,” Math. Ann., 264, 81–90 (1983).MathSciNetCrossRefGoogle Scholar
  202. 202.
    G. Schumacher, “Moduli of polarized Kähler manifolds,” Math. Ann., 269, 137–144 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  203. 203.
    G. Schumacher, “On the geometry of moduli spaces,” Manuscr. Math., 50, Nos. 1–3, 229–267 (1985).zbMATHMathSciNetCrossRefGoogle Scholar
  204. 204.
    G. Schumacher, “Harmonic maps of the moduli space of compact Riemann surfaces,” Math. Ann., 275, 466 (1986).MathSciNetCrossRefGoogle Scholar
  205. 205.
    G. Schumacher, “The theory of Teichmüller spaces. A view towards moduli spaces of Kähler manifolds,” Encycl. Math. Sci., 69, 251–310 (1990).MathSciNetGoogle Scholar
  206. 206.
    G. Schumacher, “Moduli spaces of compact Kähler manifolds. The generalized Petersson-Weil metric and positive line bundles on moduli spaces,” Rev. Roumaine Math. Pures Appl., 36, Nos. 5–6, 291–308 (1991).MathSciNetzbMATHGoogle Scholar
  207. 207.
    K. Sekigawa, “On some compact Einstein almost Kähler manifolds,” J. Math. Soc. Japan, 39, 677–684 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  208. 208.
    Shen Ying, “A note on Fischer-Marsden conjecture,” Proc. Amer. Math. Soc., 125, No. 3, 901–905 (1997).zbMATHMathSciNetCrossRefGoogle Scholar
  209. 209.
    S. Simanca, “A note on extremal metrics of nonconstant scalar curvature,” Israel. J. Math., 78, 85–93 (1992).zbMATHMathSciNetCrossRefGoogle Scholar
  210. 210.
    S. Simanca, Heat flows for extremal Kähler metrics, Preprint mathDG/0310363 (2003), http: xxx.lanl.gov.Google Scholar
  211. 211.
    N. K. Smolentsev, “On the Maupertuis principle,” Sib. Mat. Zh., 20, No. 5, 1092–1098 (1979).MathSciNetGoogle Scholar
  212. 212.
    N. K. Smolentsev, “Flow integrals of an ideal barotropic fliud,” Sib. Mat. Zh., 23, No. 1, 205–208 (1982).zbMATHMathSciNetGoogle Scholar
  213. 213.
    N. K. Smolentsev, “On the space of K-contact metrics of a three-dimensional manifold,” Sib. Mat. Zh., 28, No. 6, 119–125 (1987).zbMATHMathSciNetGoogle Scholar
  214. 214.
    N. K. Smolentsev, “Orthogonal decompositions of the space of symmetric tensors on an almost Kählerian manifold,” Sib. Mat. Zh., 30, No. 3, 131–139 (1989).zbMATHMathSciNetGoogle Scholar
  215. 215.
    N. K. Smolentsev, “On the space of associated metrics on a regular contact manifold,” Sib. Mat. Zh., 31, No. 3, 176–185 (1990).MathSciNetGoogle Scholar
  216. 216.
    N. K. Smolentsev, “On the curvature of the space of associated metrics on a symplectic manifold,” Sib. Mat. Zh., 33, No. 1, 132–139 (1992).zbMATHMathSciNetGoogle Scholar
  217. 217.
    N. K. Smolentsev, “On the curvature of the space of associated metrics on a contact manifold,” Sib. Mat. Zh., 33, No. 6, 188–194 (1992).zbMATHMathSciNetGoogle Scholar
  218. 218.
    N. K. Smolentsev, “Natural weak Riemannian structures on the space of Riemannian metrics,” Sib. Mat. Zh., 35, No. 2, 439–445 (1994).MathSciNetGoogle Scholar
  219. 219.
    N. K. Smolentsev, “Critical associated metrics on a symplectic manifold,” Sib. Mat. Zh., 36, No. 2, 359–367 (1995).zbMATHMathSciNetCrossRefGoogle Scholar
  220. 220.
    N. K. Smolentsev, “On the space of Riemannian metrics on symplectic and contact manifolds,” Sib. Mat. Zh., 42, No. 6, 1402–1407 (2001).MathSciNetGoogle Scholar
  221. 221.
    N. K. Smolentsev, “On the spaces of associated metrics on the sphere and torus,” Vestn. Kemerovo Univ., Ser. Mat., 4, 237–245 (2000).Google Scholar
  222. 222.
    N. K. Smolentsev, The space of associated metrics on a symplectic manifold, Preprint mathDG/0108110 (2001), http:xxx.lanl.gov.Google Scholar
  223. 223.
    T. N. Subramanian, Slices for the actions of smooth tame Lie groups, Thesis (1974).Google Scholar
  224. 224.
    T. N. Subramanian, “Slices for actions of infinite-dimensional groups,” Contemp. Math., 54, 65–77 (1986).Google Scholar
  225. 225.
    R. C. Swanson and C. C. Chicone, “Equivalence and slice theory for symplectic forms on closed manifolds,” Proc. Amer. Math. Soc., 73, No. 2, 265–270 (1979).zbMATHMathSciNetCrossRefGoogle Scholar
  226. 226.
    S. T. Swift, “Natural bundles, I. A minimal resolution of superspace,” J. Math. Phys., 33, 3723–3730 (1992).zbMATHMathSciNetCrossRefGoogle Scholar
  227. 227.
    S. Tanno, “Variational problems on contact Riemannian manifolds,” Trans. Amer. Math. Soc., 314, No. 1, 349–379 (1989).zbMATHMathSciNetCrossRefGoogle Scholar
  228. 228.
    V. V. Trofimov and A. T. Fomenko, “Riemannian geometry,” J. Math. Sci., 109, No. 2, 1345–1501 (2002).zbMATHMathSciNetCrossRefGoogle Scholar
  229. 229.
    A. J. Tromba, “On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric,” Manuscr. Math., 56, No. 4, 475–497 (1986).zbMATHMathSciNetCrossRefGoogle Scholar
  230. 230.
    A. J. Tromba, “On an energy function for the Weil-Petersson metric on Teichmüller space,” Manuscr. Math., 59, 249–260 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  231. 231.
    I. Vaisman, “On some variational problems for two-dimensional Hermitian metrics,” Ann. Global Anal. Geom., 8, No. 2, 137–145 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  232. 232.
    H. C. Wang, “Closed manifolds with homogeneous complex structure,” Amer. J. Math., 76, 1–32 (1954).zbMATHMathSciNetCrossRefGoogle Scholar
  233. 233.
    M. Wang, “Some examples of homogeneous Einstein manifolds in dimension seven,” Duke Math. J., 49, 23–28 (1982).zbMATHMathSciNetCrossRefGoogle Scholar
  234. 234.
    M. Wang, “Einstein metrics from symmetry and bundle constructions,” in: Surv. Differ. Geom., Vol. 6, Essays on Einstein Manifolds (C. LeBrun and M. Wang, eds.), International Press (1999), pp. 287–325.Google Scholar
  235. 235.
    M. Wang and W. Ziller, “On the isotropy representation of a symmetric space,” in: Proc. Conf. “Differential Geometry on Homogeneous Spaces,” Torino (1983), pp. 253–261.Google Scholar
  236. 236.
    M. Wang and W. Ziller, “On normal homogeneous Einstein manifolds,” Ann. Sci. Ecole Norm. Super., 18, 563–633 (1985).zbMATHMathSciNetGoogle Scholar
  237. 237.
    M. Wang and W. Ziller, “Existence and non-existence of homogeneous Einstein metrics,” Invent. Math., 84, 177–194 (1986).zbMATHMathSciNetCrossRefGoogle Scholar
  238. 238.
    M. Wang and W. Ziller, “Einstein metrics on principal bundles,” J. Differ. Geom., 31, 215–248 (1990).zbMATHMathSciNetGoogle Scholar
  239. 239.
    M. Wang and W. Ziller, “On isotropy irreducible Riemannian manifolds,” Acta Math., 166, 223–261 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  240. 240.
    J. A. Wolf, “The geometry and structure of isotropy irreducible homogeneous spaces,” Acta Math., 120, 59–148 (1968); Erratum, Acta Math., 152, 141–142 (1984).zbMATHMathSciNetCrossRefGoogle Scholar
  241. 241.
    J. A. Wolf, “Local and global equivalence for flat affine manifolds with parallel geometric structures,” Geom. Dedicata, 2, 127–132 (1973).zbMATHCrossRefGoogle Scholar
  242. 242.
    J. A. Wolf, Spaces of Constant Curvature, Publish or Perish, Boston, Massachusetts (1974).zbMATHGoogle Scholar
  243. 243.
    X. Xu, “On the existence of extremal metrics,” Pac. J. Math., 174, 555–568 (1996).zbMATHGoogle Scholar
  244. 244.
    S. Yamada, “Weil-Petersson convexity of the energy functional on classical and universal Teichmüller spaces,” J. Differ. Geom., 51, 35–96 (1999).zbMATHGoogle Scholar
  245. 245.
    S. Yamaguchi and G. Chuman, “Critical Riemannian metrics on Sasakian manifolds,” Kodai Math. J., 6, 1–13 (1983).zbMATHMathSciNetCrossRefGoogle Scholar
  246. 246.
    D. Yang, “Existence and regularity of energy-minimizing Riemannian metrics,” Int. Math. Res. Notes, 2, 7–13 (1991).CrossRefGoogle Scholar
  247. 247.
    J. W. York, “Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity,” J. Math. Phys., 14, 456–464 (1973).zbMATHMathSciNetCrossRefGoogle Scholar
  248. 248.
    J. W. York, “Covariant decompositions of symmetric tensors in the theory of gravitation,” Ann. Inst. H. Poincaré, 21, 319–332 (1974).MathSciNetGoogle Scholar
  249. 249.
    W. Ziller, “Homogeneous Einstein metrics on spheres and projective spaces,” Math. Ann., 259, 351–358 (1982).zbMATHMathSciNetCrossRefGoogle Scholar

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