Description
Book SynopsisThe totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the first eigenvalue, the Lichnerowicz-Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne-Pólya-Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.
Table of ContentsFundamentals of Riemannian Geometry; The Space of Riemannian Metrics and Continuity of the Eigenvalues; Cheeger and Yau's Estimate of the First Eigenvalue; Cheng's Estimate of the kth Eigenvalue, and Lichnerowicz and Obata's Theorem; Payne-Polya-Weinberger Type Estimate of the Dirichlet Eigenvalue; The Heat Equation and the Totality of All Closed Geodesics; The Spectral Rigidity Due to V Guillemin and D Kazhdan;
