Spectral geometry of symmetric spaces

Author:
Peter B. Gilkey

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

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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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DOI:
https://doi.org/10.1090/S0002-9947-1977-0423258-3

Article copyright:
© Copyright 1977
American Mathematical Society