A characterization of the product of two 3-spheres by the spectrum
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).
asymptotic expansion of Minakshisundaram-Pleijel
scientific article; zbMATH DE number 3937990