On the Structure of Locally Symmetric Manifolds

Abstract

This paper studies structural properties of locally symmetric submanifolds. One of the main result states that a locally symmetric submanifold M of R-n admits a locally symmetric tangential parametrization in an appropriately reduced ambient space. This property has its own interest and is the key element to establish, in a follow-up paper [7], that the spectral set lambda(-1) (M) := {X is an element of S-n : lambda(X) is an element of M} consisting of all n x n symmetric matrices having their eigenvalues on M, is a smooth submanifold of the space of symmetric matrices Sn. Here A(X) is the n-dimensional ordered vector of the eigenvalues of X.