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 $\mathcal{M}$ of $\mathbf{R}^n$ 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 $\lambda^{-1}(\mathcal{M}):=\{X \in\mathbf{S}^n:\lambda(X)\in\mathcal{M}\}$ consisting of all $n \times n$ symmetric matrices having their eigenvalues on $\mathcal{M}$, is a smooth submanifold of the space of symmetric matrices $\mathbf{S}^n$. Here $\lambda(X)$ is the $n$-dimensional ordered vector of the eigenvalues of $X$.