Some Topological and Algebraic Features of Symmetric Spaces
Um Salama *
Department of Mathematics, Faculty of Darb College University, Saudi Arabia.
Ahmed Abd Alla
Department of Mathematics, Faculty of Darb College University, Saudi Arabia.
A. Elemam
Department of Mathematics, Faculty of Education Jazan University, Saudi Arabia.
*Author to whom correspondence should be addressed.
Abstract
In this study, we introduce some approaches, geometrical and algebraic, which help to give further understanding of symmetric spaces. Symmetric space is a very important field for understanding abstract and applied features of spaces. We have introduced Riemannian Manifold, Lie groups and Lie algebras, and some of their topological and algebraic properties, with some concentration on Lie algebras and root systems , which help classification and many applications of symmetric spaces. The paper is an attempt to explain some algebraic features of symmetric spaces and how to get some of their properties using algebraic approach, concluded with some results.
Keywords: Topological spaces, metric spaces, topological manifold, Riemannian manifold, lie groups, lie algebras, root systems, homogeneous spaces, symmetric spaces