Texts and Readings in Mathematics

Up to minor changes, these are the Notes of a series of lectures given at the Tata Institute of Fundamental Research, Mumbai. They cover some basic material on affine connections, locally or globally Riemannian and Hermitian symmetric spaces. A last chapter proves the basic theorems on maximal compact subgroups of Lie groups with finitely many connected components. Familiarity with differential manifolds and the elementary theory of Lie groups and Lie algebras is assumed
1. Preliminaries
2. Orthogonal involutive Lie algebras
3. Locally symmetric spaces
4. Riemannian symmetric spaces
5. Compact groups. Klein forms of symmetric spaces
6. Hermitian symmetric spaces
7. Maximal compact subgroups of Lie groups.