Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Spectral geometry of symmetric spaces
HTML articles powered by AMS MathViewer

by Peter B. Gilkey PDF
Trans. Amer. Math. Soc. 225 (1977), 341-353 Request permission

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.
References
  • Harold Donnelly, Symmetric Einstein spaces and spectral geometry, Indiana Univ. Math. J. 24 (1974/75), 603–606. MR 413011, DOI 10.1512/iumj.1974.24.24045
  • 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 10.2748/tmj/1178242547
  • 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
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C25, 58G99, 35P15
  • Retrieve articles in all journals with MSC: 53C25, 58G99, 35P15
Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 225 (1977), 341-353
  • MSC: Primary 53C25; Secondary 58G99, 35P15
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0423258-3
  • MathSciNet review: 0423258