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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Spectral geometry of symmetric spaces

Author: Peter B. Gilkey
Journal: Trans. Amer. Math. Soc. 225 (1977), 341-353
MSC: Primary 53C25; Secondary 58G99, 35P15
MathSciNet review: 0423258
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Abstract: Let M be a compact Riemannian manifold without boundary. Let D be a differential operator on M. Let spec (D, M) denote the eigenvalues of D repeated according to multiplicity. Several authors have studied the extent to which the geometry of M is reflected by spec (D, M) for certain natural operators D. We consider operators D which are convex combinations of the ordinary Laplacian and the Bochner or reduced Laplacian acting on the space of smooth functions and the space of smooth one forms. We prove that is is possible to determine if M is a local symmetric space from its spectrum. If the Ricci tensor is parallel transported, the eigenvalues of the Ricci tensor are spectral invariants of M.

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  • Harold Donnelly, Symmetric Einstein spaces and spectral geometry, Indiana Univ. Math. J. 24 (1974/75), 603–606. MR 413011, DOI
  • Peter B. Gilkey, The spectral geometry of real and complex manifolds, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 265–280. MR 0388466
  • Peter B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differential Geometry 10 (1975), no. 4, 601–618. MR 400315
  • V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. 34 (1970), no. 3-4, 269–285 (1971). MR 0488181
  • Takashi Sakai, On eigen-values of Laplacian and curvature of Riemannian manifold, Tohoku Math. J. (2) 23 (1971), 589–603. MR 303465, DOI
  • R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943
  • Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158

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Article copyright: © Copyright 1977 American Mathematical Society