On the Structure of Locally Symmetric Manifolds

This paper studies structural properties of locally symmetric submanifolds. One of the main result states that a locally symmetric submanifold

of

\mathbf{R}^n 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 of the authors [Spectral (isotropic) manifolds and their dimension, J. Anal. Math., to appear], that the spectral set

\lambda^{-1}(\mathcal{M}):=\{X \in\mathbf{S}^n:\lambda(X)\in\mathcal{M}\} λ − 1 ( M ) : = { X ∈ S n : λ ( X ) ∈ M }

consisting of all

n \times n n × n

symmetric matrices having their eigenvalues on

\mathcal{M} M

, is a smooth submanifold of the space of symmetric matrices

\mathbf{S}^n S n

. Here

\lambda(X) λ ( X )

is the

n n

-dimensional ordered vector of the eigenvalues of

X X

.