Spectral (isotropic) manifolds and their dimension

Abstract

A set of n x n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in R-n is called spectral or isotropic. In this paper, we establish that every locally symmetric C-k submanifold M of R-n gives rise to a C-k spectral manifold for k is an element of {2, 3,...,infinity,omega}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and Silhavy and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.