Locally symmetric submanifolds lift to spectral manifolds

TL;DR

This paper proves that locally symmetric smooth submanifolds can be lifted to spectral manifolds in the space of symmetric matrices, with explicit dimension formulas based on their intrinsic properties.

Contribution

It establishes a natural correspondence between locally symmetric submanifolds and spectral manifolds, providing explicit formulas for their dimensions.

Findings

Spectral manifolds are smooth submanifolds of symmetric matrices.

Dimension formulas relate spectral manifolds to their originating locally symmetric manifolds.

The work bridges geometric properties of submanifolds with spectral matrix analysis.

Abstract

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of the locally symmetric manifold.