Some Trends in Riemannian Geometry

  • Arthur L. Besse


The sessions “Differentialgeometrie im Großen”, initiated by Professor Wilhelm Klingenberg and Professor Shiing Shen Chern, have brought to Oberwolfach a very international crowd of geometers for many years. We begin this review by presenting some highlights of talks given there in the last ten years. In the next paragraphs, we selected four lines of development of the field that we tried to put in perspective, namely, Isospectrality, Dynamical Geometry, Deforming metrics, and Moduli spaces of geometric objects. The influence of other areas of Mathematics, and also of Physics, will often be perceivable.


Modulus Space Riemannian Manifold Riemannian Geometry Closed Geodesic 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.
    Abresch, U.: Constant mean curvature tori in terms of elliptic functions. J. Reine Ang. Math. 374 (1987) 169–192MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Anderson, M.: The L 2 -structure of moduli spaces of Einstein metrics on 4-manifolds. Geom. Functional Anal. 2 (1992) 29–89MATHCrossRefGoogle Scholar
  3. 3.
    Atiyah, M.F.: Geometry of Yang-Mills fields. Lezioni Fermiane. Acad. Naz. dei Lincei, Sc. Norm. Sup. Pisa, 1979Google Scholar
  4. 4.
    Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London 362 (1978) 425–461MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bando, S.: On 3-dimensional compact Kähler manifolds of non-negative bisectional curvature. J. Differential Geom. 19 (1984) 283–297MATHMathSciNetGoogle Scholar
  6. 6.
    Bemelmans, J., Min, Oo, M., Ruh, E.: Smoothing Riemannian metrics. Math. Z. 188 (1984) 69–74MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Benoist, Y., Foulon, P., Labourie, F.: Flots d’ Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5 (1992) 33–74MATHMathSciNetGoogle Scholar
  8. 8.
    Bérard, P.: Variétés riemanniennes isospectrales non isométriques. Astérisque 177–178 (1989) 127–154Google Scholar
  9. 9.
    Bérard, P.: Transplantation et isospectralité I. Math. Ann. 292 (1992) 547–559MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Besse, A.L.: Manifolds all of whose geodesics are closed. Ergebnisse der Mathemathik, vol. 98. Springer, Berlin 1973Google Scholar
  11. 11.
    Besse, A.L.: Einstein manifolds. Ergebnisse der Mathematik, vol. 10. Springer, Berlin Heidelberg New York 1987Google Scholar
  12. 12.
    Besson, G.: On the multiplicity of the eigenvalues of the Laplacian. In: Geometry and analysis on manifolds, Katata/Kyoto. Lecture Notes in Mathematics, vol. 1339 (1987) 32–53CrossRefGoogle Scholar
  13. 13.
    Besson, G., Courtois, G., Gallot, S.: Volume minimal des espaces localement symétriques. Inventiones Math. 103 (1991) 417–445MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Besson, G., Courtois, G., Gallot, S.: Les variétés hyperboliques sont des minima locaux de l’entropie topologique. lnventiones Math. (to appear)Google Scholar
  15. 15.
    Bourguignon, J.P.: L’équation de la chaleur associée à la courbure de Ricci, d’après R.S. Hamilton. In: Séminaire Bourbaki 1985–1986, Exposé 653. Astérisque 145—146 (1987) 45–62Google Scholar
  16. 16.
    Bowen, R.: Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972) 1–30MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Brooks, R.: The spectral geometry of a tower of coverings. J. Differential Geom. 23 (1986) 97–107MATHMathSciNetGoogle Scholar
  18. 18.
    Brooks, R.: Constructing isospectral manifolds. Amer. Math. Monthly 95 (1988) 823–839MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Burger, M.: Spectre du laplacien, graphes et topologie de Fell. Comment. Math. Helv. 63 (1988) 226–252MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Burns, K., Gerber, M.: Real analytic Bernoulli geodesic flows on S 2 . J. Ergodic Th. Dynamical Systems 9 (1989) 27–45MATHMathSciNetGoogle Scholar
  21. 21.
    Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Bâle 1992MATHGoogle Scholar
  22. 22.
    Cao, H.D.: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Inventiones Math. 81 (1985) 359–372MATHCrossRefGoogle Scholar
  23. 23.
    Cao, H.D.: Compact Kähler manifolds with non-negative curvature operator. Inventiones Math. 83 (1986) 553–556MATHCrossRefGoogle Scholar
  24. 24.
    Chow, B.: The Ricci flow on the 2-sphere. J. Differential Geom. 33 (1991) 325–334MATHMathSciNetGoogle Scholar
  25. 25.
    Colin de Verdière, Y.: Spectre du laplacien et longueurs des géodésiques périodiques I. Compositio Math. 27 (1973) 83–106; idem: II, ibidem, 159–184MATHMathSciNetGoogle Scholar
  26. 26.
    Colin de Verdière, Y.: Sur la multiplicité de la première valeur propre non nulle du laplacien. Comment. Math. Helv. 61 (1986) 254–270MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Colin de Verdière, Y.: Construction de laplaciens dont une partie finie du spectre est donnée. Ann. Sci. Ecole Norm. Sup. Paris 20 (1987) 599–615MATHGoogle Scholar
  28. 28.
    DeTurck, D.: Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18 (1983) 157–162MathSciNetGoogle Scholar
  29. 29.
    DeTurck, D., Gordon, C.: Isospectral deformations I: Riemannian structures on two-step nilspaces. Commun. Pure Appl. Math. 40 (1987) 367–387, idem: II, Trace formulas, metrics, and potentials, ibidem 42 (1989) 1067–1095CrossRefGoogle Scholar
  30. 30.
    DeTurck, D., Gluck, H., Gordon, C., Webb, D.: The inaudible geometry of nilmanifolds. Inventiones Math. 111 (1992) 271–284CrossRefGoogle Scholar
  31. 31.
    Donaldson, S.K.: An application of gauge theory to the topology of 4-manifolds. J. Differential Geom. 18 (1983) 279–315MATHMathSciNetGoogle Scholar
  32. 32.
    Donaldson, S.K.: Connections, cohomology and the intersection form of 4-manifolds. J. Differential Geom. 24 (1986) 275–342MATHMathSciNetGoogle Scholar
  33. 33.
    Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. Oxford Univ. Press, Oxford, 1990MATHGoogle Scholar
  34. 34.
    Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Inventiones Math. 29 (1975) 39–79MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Eberlein, P.: When is a geodesic flow of Anosov type? I. J. Differential Geom. 8 (1973) 437–463MATHMathSciNetGoogle Scholar
  36. 36.
    Foulon, P., Labourie, F.: Sur les variétés compactes asymptotiquement harmoniques. Inventiones Math. 109 (1992) 97–112MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Freed, D.S., Uhlenbeck, K.K.: Instantons and four-manifolds. Math. Sci. Res. Inst. Publications 1 (1984) Springer, Berlin Heidelberg New YorkGoogle Scholar
  38. 38.
    Gao, L.Z.: Convergence of Riemannian metrics: Ricci and L n/2-pinching. J. Differential Geom. 32 (1990) 349–382MATHMathSciNetGoogle Scholar
  39. 39.
    Gauduchon, P.: Variétés riemanniennes autoduales. Séminaire Bourbaki, Exposé 767 (1993) 1–27Google Scholar
  40. 40.
    Ghys, E.: Rigidité différentiable des groupes fuchsiens. Preprint, Ec. Norm. Sup. Lyon, 1992Google Scholar
  41. 41.
    Gordon, C., Wilson, E.: Isospectral deformations of compact solvmanifolds. J. Differential Geom. 19 (1984) 241–256MATHMathSciNetGoogle Scholar
  42. 42.
    Gordon, C., Webb, D., Wolpert, S.: Isospectral plane domains and surfaces via Riemannian orbifolds. Inventiones Math. 110 (1992) 1–22MATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Green, L.: Surfaces without conjugate points. Trans. Amer. Math. Soc. 76 (1954) 529–546MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Groisser, D., Parker, T.: The Riemannian geometry of the Yang-Mills moduli space. Commun. Math. Phys. 112 (1987) 663–689MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Groisser, D., Parker, T.: The geometry of the Yang-Mills moduli space for definite manifolds. J. Differential Geom. 29 (1989) 499–544MATHMathSciNetGoogle Scholar
  46. 46.
    Gromov, M.: Manifolds of negative curvature. J. Differential Geom. 13 (1978) 223–230MATHMathSciNetGoogle Scholar
  47. 47.
    Gromov, M.: The foliated Plateau problem: I. Geom. Functional Anal. 1 (1992) 14–79,; idem: II, ibidem, 253–320MathSciNetCrossRefGoogle Scholar
  48. 48.
    Hadamard, J.: Les surfaces à courbures opposées et leurs lignes géodésiques. J. Math. Pures Appl. 4 (1898) 27–74Google Scholar
  49. 49.
    Hamenstädt, U.: A new description of the Bowen-Margulis measure. J. Ergodic Th. Dynamical Systems 9 (1989) 455–464MATHGoogle Scholar
  50. 50.
    Hamilton, R.S.: 3-Manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982) 255–306MATHMathSciNetGoogle Scholar
  51. 51.
    Hamilton, R.S.: The Ricci flow for surfaces. Contemporary Math. 71 (1988) 237–262MathSciNetGoogle Scholar
  52. 52.
    Hamilton, R.S.: The Harnack estimate for the Ricci flow. J. Differential Geom. 37 (1993) 225–243MATHMathSciNetGoogle Scholar
  53. 53.
    Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986) 153–179MATHMathSciNetGoogle Scholar
  54. 54.
    Hitchin, N.J.: The Yang-Mills equations and the topology of 4-manifolds. In: Séminaire Bourbaki 82–83, Exposé 606. Astérisque 105–106 (1983) 167–178MathSciNetGoogle Scholar
  55. 55.
    Hitchin, N.J., Karlhede, A., Lindström, U., Rocek, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108 (1987) 535–589MATHCrossRefGoogle Scholar
  56. 56.
    Huisken, G.: Ricci deformation of the metric on a Riemannian manifold. J. Differential Geom. 17 (1985) 47–62MathSciNetGoogle Scholar
  57. 57.
    Hsiang, W.Y., Palais, R.S., Temg, C.L.: The topology of isoparametric submanifolds. J. Diftferential Geom. 27 (1988) 423–460MATHGoogle Scholar
  58. 58.
    Ikeda, A.: Riemannian manifolds p-isospectral but not (p+ 1)-isospectral. In: Geometry of manifolds. Acad. Press, New York 1989Google Scholar
  59. 59.
    Itoh, M.: Moduli ofhaif conformally flat structures. Math. Ann. (to appear)Google Scholar
  60. 60.
    Katok, A.: Four applications of conformal equivalence to geometry and dynamics. J. Ergodic Th. Dynamical Systems 8 (1988) 215–240MathSciNetCrossRefGoogle Scholar
  61. 61.
    Katok, A., Spatzier, R.: Differential rigidity of hyperbolic Abelian actions. Preprint, Math. Sci. Res. Inst., Berkeley 1992Google Scholar
  62. 62.
    King, A.D., Kotschick, D.: The deformation theory of anti-self-dual conformal structures. Math. Ann. 294 (1992) 591–609MATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Knieper, G.: Spherical means on compact Riemannian manifolds of negative curvature. Habilitationsschrift, Institut für Mathematik, Augsburg 1992Google Scholar
  64. 64.
    Koiso, N.: Rigidity and infinitesimal deformability of Einstein metrics. Osaka J. Math. 29 (1982) 643–668MathSciNetGoogle Scholar
  65. 65.
    Lawson, H.B. Jr.: The theory of gauge fields in four dimensions. C.B.M.S. Regional Conference Series 58 (1985)Google Scholar
  66. 66.
    Ledrappier, F.: Harmonic measures and Bowen-Margulis measures. Israel J. Math. 71 (1990) 275–288MATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Lyons, T., Sullivan, D.: Function theory, random paths and covering spaces. J. Differential Geom. 19 (1984) 299–323MATHMathSciNetGoogle Scholar
  68. 68.
    Margerin, Ch.: Pour une théorie de la déformation en géométrie métrique: Chap. I: Intégrabilité locale; Chap. 2: Intégrabilité globale; Chap. 4: Un résultat optimal en dimension 4; Chap. 5: Un résultat optimal asymptotique. Preprint, Ecole Polytechnique, 1992Google Scholar
  69. 69.
    Margerin, Ch.: Pointwise-pinched manifolds are space forms. Proc. Amer. Math. Soc. Symp. Pure Math. (Arcata 1984) 44 (1986) 307–328MathSciNetGoogle Scholar
  70. 70.
    Margerin, Ch.: A sharp theorem for weakly-pinched 4-manifolds. C.R. Acad. Sci. Paris 303 (1986) 877–880MATHMathSciNetGoogle Scholar
  71. 71.
    Margerin, Ch.: Fibrés stables et métriques d’Hermite-Einstein. Séminaire Bourbaki, Exposé 683. Astérisque 152–153 (1987) 263–283MathSciNetGoogle Scholar
  72. 72.
    Margulis, G.A.: On some applications of ergodic theory to the study of manifolds of negative curvature. Funkt. Anal. i Prilozhen. 3 (1969) 89–90MathSciNetGoogle Scholar
  73. 73.
    McKean, H.: Selberg’s trace formula as applied to a compact Riemann surface. Commun. Pure Appli. Math. 25 (1972) 225–246MathSciNetCrossRefGoogle Scholar
  74. 74.
    Min Oo, M., Ruh, E.: Curvature deformations. In: Curvature and topology of Riemannian manifolds. Lectures Notes in Mathematics, vol. 1201. Springer (1986) 180–190Google Scholar
  75. 75.
    Min Oo, M., Ruh, E.: L 2-curvature pinching. Comment. Math. Helvetici 65 (1990) 36–51MATHCrossRefGoogle Scholar
  76. 76.
    Mok, N.: The uniformization theorem for compact Kähler manifolds of non-negative holomorphic bisectional curvature. J. Differential Geom. 27 (1988) 179–214MATHMathSciNetGoogle Scholar
  77. 77.
    Nadirashvili, N.S.: Multiple eigenvalues of the Laplace operator. Math. USSR Sbornik 61 (1988) 225–238MathSciNetCrossRefGoogle Scholar
  78. 78.
    Nishikawa, S.: Deformation of Riemannian manifolds. Proc. Amer. Math. Soc. Symp. Pure Math. (Arcata 1984) 44 (1986) 343–352MathSciNetGoogle Scholar
  79. 79.
    Osgood, B., Phillips, R., Sarnak, P.: Compact isospectral sets of surfaces. J. Functional Anal. 80 (1988) 212–234MATHMathSciNetCrossRefGoogle Scholar
  80. 80.
    Ouyang, H., Pesce, H.: Déformations isospectrales sur les nilvariétés de rang deux. C.R. Acad. Sci. Paris 314 (1992) 621–623MATHMathSciNetGoogle Scholar
  81. 81.
    Pansu, P.: Le flot géodésique des variétés riemanniennes à courbure négative. Séminaire Bourbaki, Exp. 738, Astérisque 202 (1991) 269–298MathSciNetGoogle Scholar
  82. 82.
    Penner, R.C.: Calculus on moduli spaces. In: Geometry of group representations. Contemporary Math., Amer. Math. Soc., Providence, 74 (1988) 277–293MathSciNetGoogle Scholar
  83. 83.
    Prat, J.J.: Etude et convergence angulaire du mouvement brownien sur une variété à courbure négative. C.R. Acad. Sci. Paris 290 (1975) 1539–1542MathSciNetGoogle Scholar
  84. 84.
    Séminaire Palaiseau: Géométrie des surfaces K3: Modules et périodes. Astérisque 126, 1983Google Scholar
  85. 85.
    Shi, W.X.: Deforming the metric on complete Riemannian manifolds. J. Differential Geom. 30 (1989) 223–301MATHMathSciNetGoogle Scholar
  86. 86.
    Simpson, C.T.: Constructing variations of Hodge structures using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1 (1988) 867–918MATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. Math. 121 (1985) 169–186MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Sunada, T.: Fundamental groups and laplacians. In: Geometry and analysis on manifolds Katata/Kyoto. Lecture Notes in Mathematics, vol. 1339. Springer (1987) 248–277CrossRefGoogle Scholar
  89. 89.
    Taubes, C.H.: Self-dual Yang-Mills connections on non selfdual 4-manifolds. J. Differential Geom. 17 (1982) 139–170MATHMathSciNetGoogle Scholar
  90. 90.
    Taubes, C.H.: Self-dual connections on manifolds with indefinite intersection matrix. J. Differential Geom. 19 (1984) 345–391Google Scholar
  91. 91.
    Taubes, C.H.: Long range forces and topology ofinstanton moduli spaces. In: Colloque en l’honneur de Laurent Schwartz. Astérisque 132 (1985) 243–255Google Scholar
  92. 92.
    Taubes, C.H.: The stable topology of moduli spaces. J. Differential Geom. 29 (1989) 163–230MATHMathSciNetGoogle Scholar
  93. 93.
    Taubes, C.H.: Casson’s invariant and gauge theory. J. Differential Geom. 31 (1990) 547–599MATHMathSciNetGoogle Scholar
  94. 94.
    Taubes, C.H.: The existence of anti-self-dual conformal structures. J. Differential Geom. 36 (1992) 163–253MATHMathSciNetGoogle Scholar
  95. 95.
    Thorbergsson, G.: Isoparametric foliations and their buildings. Ann. Math. 133 (1991) 429–446MATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    Tian, G., Yau, S.T.: Kähler-Einstein metrics on complex surfaces with c 1 > 0. Commun. Math. Phys. 42 (1987) 175–203MathSciNetCrossRefGoogle Scholar
  97. 97.
    Tromba, A.J.: Teichmüller theory in Riemannian geometry. Lect. in Math. ETH Zürich. Birkhäuser, Basel 1992Google Scholar
  98. 98.
    Wolpert, S.: The eigenvalue spectrum as moduli for compact Riemann surfaces. Bull. Amer. Math. Soc. 83 (1977) 1306–1308MATHMathSciNetCrossRefGoogle Scholar
  99. 99.
    Yang, D.: L p-pinching and compactness theorems for compact Riemannian manifolds on curvature: I. Ann. Sci. Ecole Norm. Sup. Paris 25 (1992) 77–105; idem: II, ibidem, 179–199MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Arthur L. Besse

There are no affiliations available

Personalised recommendations