Symmetric Spaces and Their Applications

Abstract

Symmetric spaces is a special topic in Riemannian geometry . These spaces were first studied and classified by Elie Cartan . In this research we study in a logical ordering their stnicture through manifolds . Lie groups . Lie algebras and basics of Riemannian geometry. The study covers locally and globally symmetric spaces . endowed with some examples for them such as Euclidean spaces . spheres hyperbolic spaces and some applications on locally symmetric spaces in the field of arithmetic and algebraic groups , including quadratic and modular forms , lattices . the realization of discrete series representations of groups . Poincare and linear symmetric spaces . Their classification is also discussed by introducing compact and noncompact symmetric spaces besides the types ( l . ll . lll . l\‘ ) . This classification is carried tluough Lie algebras . root systems and their Dynkin diagrams t The aim ofthe research is to display in a simplified manner the connection between symmetric spaces and the differential geometric terms such as manifolds . Lie groups , Lie algebras and basics of Riemannian geometry with some of the applications of symmetric spaces.