The spectrum of the Laplacian for $1$-forms
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- by Shûkichi Tanno PDF
- Proc. Amer. Math. Soc. 45 (1974), 125-129 Request permission
Abstract:
Let $(M,g)$ and $(M’,g’)$ be compact orientable Riemannian manifolds with the same spectrum of the Laplacian for $1$-forms. We prove that, for $\dim M = 2,3,16,17, \cdots ,93,(M,g)$ is of constant curvature if and only if $(M’,g’)$ is so.References
- M. Berger, Le spectre des variétés riemanniennes, Rev. Roumaine Math. Pures Appl. 13 (1968), 915–931 (French). MR 239535
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
- Matthew P. Gaffney, Asymptotic distributions associated with the Laplacian for forms, Comm. Pure Appl. Math. 11 (1958), 535–545. MR 99541, DOI 10.1002/cpa.3160110405
- V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geometry 5 (1971), 233–249. MR 292114
- 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
- Shǔkichi Tanno, An inequality for $4$-dimensional Kählerian manifolds, Proc. Japan Acad. 49 (1973), 257–261. MR 372796
- Shǔkichi Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tǒhoku Math. J. (2) 25 (1973), 391–403. MR 0334086, DOI 10.2748/tmj/1178241341
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 125-129
- MSC: Primary 58G15; Secondary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-1974-0343321-8
- MathSciNet review: 0343321