A characterization of the product of two 3-spheres by the spectrum
type: Item
instance of: scholarly article
title: A characterization of the product of two 3-spheres by the spectrum
zbMATH Open document ID: 0585.53034
author: Shukichi Tanno
author: Kazuo Masuda
published in: Kodai Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let M be a compact Riemannian manifold and Spec (M) the spectrum of the Laplacian acting on functions on M. An interesting problem is to investigate relations between Spec (M) and Riemannian structures. The authors prove the following theorem: Let M, M' be 3-dimensional compact Riemannian manifolds. If \(Spec (M\times M')=Spec (S^ 3(c)\times S^ 3(c')),\) then M, M' are of constant sectional curvature k and k' with \(k+k'=c+c'\); moreover if k and k' are positive, then M is isometric to \(S^ 3(c)\) (or \(S^ 3(c'))\) and M' is isometric to \(S^ 3(c')\) (or \(S^ 3(c)).\) The authors prove another version of this theorem, in which the Euclidean 3-sphere \(S^ 3(c)\) is replaced by the 3-dimensional complex projective space. In order to prove these theorems the authors use the coefficients of the asymptotic expansion of Minakshisundaram-Pleijel for Spec (M).
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 3937990
zbMATH Keywords: isospectral
zbMATH Keywords: spectrum of the Laplacian
zbMATH Keywords: constant sectional curvature
zbMATH Keywords: asymptotic expansion of Minakshisundaram-Pleijel
reviewed by: Domenico Perrone
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138037106
OpenAlex ID: W1975953985
cites work: Q5573057
cites work: Curvature and the eigenvalues of the Laplacian
cites work: Q5647591
cites work: On eigen-values of Laplacian and curvature of Riemannian manifold
cites work: Eigenvalues of the Laplacian of Riemannian manifolds
cites work: A characterization of the canonical spheres by the spectrum
cites work: A characterization of a complex projective space by the spectrum
Wikidata QID: Q125767345
DOI: 10.2996/KMJ/1138037106
description: scientific article; zbMATH DE number 3937990