ON EIGENVALUES OF THE LAPLACIAN AND CURVATURE OF RIEMANNIAN MANIFOLDS

Let (M('n),g) be a compact connected orientable Riemannian manifold of dimension n,g the fundamental tensor. For 0 (LESSTHEQ) p (LESSTHEQ) n let (DELTA) be the Laplace-Beltrami operator on exterior p-form on M. The set. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). of eigenvalues of (DELTA) is the spectrum of M. The question of how the spectrum of M determines the structure of M has been studied by various authors. It is proved that the asymptotic expansion (the heat expansion). (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). holds, where a(,i,p) are Riemannian invariants. The coefficients a(,0,0), a(,1,0), a(,2,0) have been computed by M. Berger and by McKean and Singer. a(,1,p), a(,2,p) have been computed by V. K. Patodi, and a(,3,0) by T. Sakai.(,). In this dissertation, for various special kinds of manifolds, we write out these coefficients and the coefficients b(,i)(i=1,2,3) of. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). the power series expansion (in normal coordinates) for the volume of a small geodesic ball with center m and radius r computed by A. Gray and L. Vanhecke. By forming tables for these coefficients we find several relations among them for some of the manifolds. We then compute the coefficients a(,3,p) and a(,4,p) for 2-dimensional compact Riemannian manifolds.