Dimension, volume, and spectrum of a Riemannian manifold
It is well known that the spectrum of the Laplace operator defined on a Riemannian manifold determines certain aspects of its geometry. For instance, Weyl's formula shows that the knowledge of the whole spectrum determines both the volume and the dimension of the manifold. In this paper the author studies the following related problem: how many eigenvalues of the Laplace operator it takes to determine the dimension? The following answer is obtained: If \(M\) is a connected compact Riemannian manifold without boundary such that the sectional curvature is bounded from above by a positive number and bounded from below. Then there exists a natural number \(N\), depending only on the bounds of the sectional curvature, the injectivity radius and the first eigenvalue, such that the dimension of \(M\) is determined by the first \(N\) eigenvalues. A counterexample is given showing that there is no finite number of eigenvalues that suffices to determine the dimension of a Riemannian manifold uniformly for the class of all Riemannian manifolds. An easier answer is obtained for connected convex Euclidean domains. Moreover, the sample problem for the volume of the manifold is also studied.
scientific article; zbMATH DE number 561188