A hymn to the symmetric spaces (Q2779200)

This short article is a contribution of the author to the proceedings of a workshop held a decade ago in honor of M. Obata. Following the author ``it is a survey or rather a warm and informal invitation to the geometric theory of symmetric spaces along with its applications''. It is a well-known fact that the author contributed in a fundamental way to this domain of study.NEWLINENEWLINENEWLINEThe first part contains a brief but rich exposition of the geometric theory of symmetric spaces containing a list of facts about the class of connected compact ones. The author dwells on definitions, homomorphisms, \(G\)-invariant complex structures, the 2-number, polars and meridians, the signature (cohomological index), the rank, root systems, the arithmetic distance, the algebraic, analytic and topological structure of symmetric \(R\)-spaces, etc.NEWLINENEWLINENEWLINEIn the second section, he treats some more recent developments related to the study of the space \(C^\infty (M,N)\) of all smooth maps of \(M\) into \(N\) and focuses on a lot of aspects of the theory of variations: critical points, \(p\)-harmonicity, \(p\)-stability, Jacobi operators, minimality, stability for minimal submanifolds. Several open problems are mentioned.NEWLINENEWLINENEWLINEThe exposition also contains a rich list of references.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].