Acta Mathematica

, 142:157 | Cite as

Riemannian geometry as determined by the volumes of small geodesic balls

  • A. Gray
  • L. Vanhecke
Article

Keywords

Manifold Riemannian Manifold Symmetric Space Sectional Curvature Curvature Tensor 

References

  1. [1].
    Alekseevskij, D. V., On holonomy groups of Riemannian manifolds.Ukrain. Math. Z., 19 (1967), 100–104.MATHGoogle Scholar
  2. [2].
    Allendoerfer, C. B., Steiner's formula on a generalS n+1.Bull. Amer. Math. Soc., 54 (1948), 128–135.MATHMathSciNetGoogle Scholar
  3. [3].
    Berger, M., Sur les variétés d'Einstein compactes.C.R. III e Réunion Math. Expression Latine, Namur, (1965), 35–55.Google Scholar
  4. [4].
    —, Le spectre des variétés riemanniennes.Rev. Roumaine Math. Pures Appl., 23 (1968), 915–931.Google Scholar
  5. [5].
    Berger, M., Gauduchon, P. &Mazet, E.,Le spectre d'une variété riemannienne. Lecture Notes in Mathematics, vol. 194, Springer-Verlag, Berlin and New York, 1971.MATHGoogle Scholar
  6. [6].
    Bertrand, J., Diguet &Puiseux, V., Démonstration d'un théorème de Gauss.Journal de Mathématiques, 13 (1848), 80–90.Google Scholar
  7. [7].
    Besse, Arthur L.,Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik, vol. 93, Springer-Verlag, Berlin and New York, 1978.MATHGoogle Scholar
  8. [8].
    Borel, A., On the curvature operator of the Hermitian symmetric manifolds.Ann. of Math., 71 (1960), 508–521.MATHMathSciNetCrossRefGoogle Scholar
  9. [9].
    Brown, R. B. &Gray, A., Manifolds whose holonomy group is a subgroup of Spin (9).Differential Geometry (in honor of K. Yano). Kinokuniya, Tokyo, 1972, 41–59.Google Scholar
  10. [10].
    Calabi, E. &Vesentini, E., On compact locally symmetric Kähler manifolds.Ann. of Math., 71 (1960), 472–507.MATHMathSciNetCrossRefGoogle Scholar
  11. [11].
    Dessertine, J. C., Expressions nouvelles de la formule de Gauss-Bonnet en dimension 4 et 6.C.R. Acad. Sci. Paris Sér. A, 273 (1971), 164–167.MATHMathSciNetGoogle Scholar
  12. [12].
    Donelly, H., Symmetric Einstein spaces and spectral geometry.Indiana Univ. Math. J., 24 (1974), 603–606.MathSciNetCrossRefGoogle Scholar
  13. [13].
    —, Topology and Einstein Kaehler metrics.J. Differential Geometry, 11, (1976), 259–264.MathSciNetGoogle Scholar
  14. [14].
    Eisenhart, L. P.,A treatise on the differential geometry of curves and surfaces. Ginn, 1909.Google Scholar
  15. [15].
    Erbacher, J., Riemannian manifolds of constant curvature and the growth function of submanifolds.Michigan Math. J., 19 (1972) 215–223.MATHMathSciNetCrossRefGoogle Scholar
  16. [16].
    Gauss, C. F., Disquisitiones generales circa superficies curvas.C. F. Gauss Werke Band IV, Georg Olm Verlag, Hildesheim, 1973, 219–258.Google Scholar
  17. [17].
    Gilkey, P., The spectral geometry of a Riemannian manifold.J. Differential Geometry, 10 (1975), 601–618.MATHMathSciNetGoogle Scholar
  18. [18].
    —, Local invariants of the Riemannian metric for two-dimensional manifolds.Indiana Univ. Math. J., 23 (1974), 855–882.MATHMathSciNetCrossRefGoogle Scholar
  19. [19].
    Gray, A., Invariants of curvature operators of four-dimensional Riemannian manifolds.Canadian Math. Congress Proc. of the 13th Biennial Seminar (1971), vol. 2, 42–65.Google Scholar
  20. [20].
    —, The volume of a small geodesic ball of a Riemannian manifold.Michigan Math. J., 20 (1973), 329–344.MATHMathSciNetGoogle Scholar
  21. [21].
    —, Geodesic balls in Riemannian product manifolds.Differential Geometry and Relativity (in honor of A. Lichnerowicz), Reidel Publ. Co., Dordrecht, 1976, 63–66.Google Scholar
  22. [22].
    —, Weak holonomy groups.Math. Z., 123 (1971), 290–300.MATHMathSciNetCrossRefGoogle Scholar
  23. [23].
    —, Some relations between curvature and characteristic classes.Math. Ann., 184 (1970), 257–267.MATHMathSciNetCrossRefGoogle Scholar
  24. [24].
    —, A generalization of F. Schur's theorem.J. Math. Soc. Japan, 21 (1969), 454–457.MATHMathSciNetCrossRefGoogle Scholar
  25. [25].
    —, Pseudo-Riemannian almost product manifolds and submersionsJ. Math. Mech., 16 (1967), 715–738.MATHMathSciNetGoogle Scholar
  26. [26].
    Grossman, N., On characterization of Riemannian manifolds by growth of tubular neighborhoods.Proc. Amer. Math. Soc., 32 (1972), 556–560.MATHMathSciNetCrossRefGoogle Scholar
  27. [27].
    Günther, P., Einige Sätze über das Volumenelement eines Riemannschen Raumes.Publ. Math. Debrecen, 7 (1960), 78–93.MATHMathSciNetGoogle Scholar
  28. [28].
    Holzsager, R. A., Riemannian manifolds of finite order.Bull. Amer. Math. Soc., 78 (1972), 200–201.MATHMathSciNetCrossRefGoogle Scholar
  29. [29].
    —, A characterization of Riemannian manifolds of constant curvature.J. Differential Geometry, 8 (1973), 103–106.MATHMathSciNetGoogle Scholar
  30. [30].
    Holzsager, R. A. &Wu, H., A characterization of two-dimensional Riemannian manifolds of constant curvature.Michigan Math. J., 17 (1970), 297–299.MATHMathSciNetCrossRefGoogle Scholar
  31. [31].
    Hotelling, H., Tubes and spheres inn-spaces, and a class of statistical problems.Amer. J. Math., 61 (1939), 440–460.MATHMathSciNetCrossRefGoogle Scholar
  32. [32].
    Liouville, J., Sur un théorème de M. Gauss concernant le produit des deux rayons de courbure principaux en chaque point d'une surface.Journal de Mathématiques, 12 (1847), 291–304.Google Scholar
  33. [33].
    Sakai, T., On eigen-values of Laplacian and curvature of Riemannian manifolds.Tôhoku Math. J., 23, (1971), 589–603.MATHGoogle Scholar
  34. [34].
    Singer, I. M. &Thorpe, J. A., The curvature of 4-dimensional Einstein spaces.Global Analysis (papers in honor of K. Kodaira), Univ. of Tokyo Press, Tokyo, 1969, 355–365.Google Scholar
  35. [35].
    Vermeil, H., Notiz über das mittlere Krümmungsmass einern-fach ausgedehnten Riemann'sche Mannigfaltigkeit.Akad. Wiss. Göttingen Nachr., (1917), 334–344.Google Scholar
  36. [36].
    Watanabe, Y., On the characteristic functions of harmonic quaternion Kählerian spaces.Ködai Math. Sem. Rep., 27 (1976), 410–420.MATHGoogle Scholar
  37. [37].
    Weyl, H., On the volume of tubes.Amer. J. Math., 61 (1939), 461–472.MATHMathSciNetCrossRefGoogle Scholar
  38. [38].
    Wu, H., A characteristic property of the Euclidean plane.Michigan Math. J., 16 (1969), 141–148.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Almqvist & Wiksell 1979

Authors and Affiliations

  • A. Gray
    • 1
  • L. Vanhecke
    • 2
  1. 1.University of MarylandCollege ParkUSA
  2. 2.Katholieke Universiteit LeuvenLeuvenBelgium

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