Abstract

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

Author information

Contact details are reproduced from the original publication and may be historical.

Aris Daniilidis

Departament de Matemàtiques C1/308, Universitat Autònoma de Barcelona, 08193 Bellaterra - Cerdanyola del Vallès, Spain
and: DIM-CMM, Universidad de Chile, Blanco Encalada 2120, piso 5, Santiago, Chile

arisd@mat.uab.cat

Hristo Sendov

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

hssendov@stats.uwo.ca

Keywords

  • Locally symmetric manifold
  • spectral manifold
  • permutation
  • partition
  • symmetric matrix
  • eigenvalue

Mathematics Subject Classification

15A18, 53B25; 47A75, 05A05

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Suggested citation

A. Daniilidis, J. Malick, H. Sendov. “On the Structure of Locally Symmetric Manifolds.” Journal of Convex Analysis 22 (2015), No. 2, 399–426. https://doi.org/10.68381/jca22018

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.