On the spectral rigidity of 𝐶𝑃ⁿ

Perrone, Domenico

doi:10.1090/S0002-9939-1988-0964867-5

On the spectral rigidity of $\textbf {C}\textrm {P}^ n$
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Proc. Amer. Math. Soc. 104 (1988), 871-875 Request permission

Abstract:

Complex projective space $C{P^n}$ with the Fubini-Study metric has recently been characterized by the spectrum of the Laplacian on $2$-forms. This important result was proved separately for $n \ne 2,8$ by B. Y. Chen and L. Vanhecke, and for $n = 2$ and $n = 8$ by S. I. Goldberg. In this paper, we give a new proof which does not distinguish the three cases. It makes strong use of a result of S. Kobayashi and T. Ochiai, and may be applied to the spectrum of the Laplacian on $1$-forms. Moreover, a characterization of $C{P^2}$ by the spectrum of the Laplacian on $1$-forms is given.

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