Eigenvalue functions of symmetric tensorfields on Riemannian manifolds

Abstract

It is well-known that eigenvalues of a symmetric linear operator on an n-dimensional real vector space are all real. In general, for a symmetric tensor field of type (1, 1) on a connected n-dimensional Riemannian manifold M, the eigenvalues at each point formed n real-valued functions. These eigenvalue functions are continuous but not necessarily differentiable on the whole of M. In the talk, we shall first discuss the differentiability of the eigenvalue functions and then discuss some of the applications to the Riemannian geometry of hypersurfaces.