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Journal of Convex Analysis 22 (2015), No. 2, 399--426
Copyright Heldermann Verlag 2015

On the Structure of Locally Symmetric Manifolds

Aris Daniilidis
Departament de Matemàtiques C1/308, Universitat Autònoma de Barcelona, 08193 Bellaterra - Cerdanyola del Vallès, Spain

Jérôme Malick
CNRS, Laboratoire J. Kunztmann, Grenoble, France

Hristo Sendov
Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada


\newcommand{\RR}{\mathbf{R}} \newcommand{\Sn}{{\bf S}^n} \newcommand{\Mm}{\mathcal{M}} This paper studies structural properties of locally symmetric submanifolds. One of the main result states that a locally symmetric submanifold $\Mm$ of $\RR^n$ 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 $\lambda^{-1}(\Mm):=\{X \in\Sn :\lambda(X)\in\Mm\}$ consisting of all $n \times n$ symmetric matrices having their eigenvalues on $\Mm$, is a smooth submanifold of the space of symmetric matrices $\Sn$. Here $\lambda(X)$ is the $n$-dimensional ordered vector of the eigenvalues of $X$.

Keywords: Locally symmetric manifold, spectral manifold, permutation, partition, symmetric matrix, eigenvalue.

MSC: 15A18, 53B25; 47A75, 05A05

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