Nemaat Hamed Taleb2018-08-012018-08-012008http://hdl.handle.net/123456789/12326A thesis submitted for the degree of Ph. D. in MathematicsABSTRACT We study manifolds spaces ‘which lead to Lie group, then by definition of transitive action of a Lie group in a manifold we get homogeneous spaces. ' Also we study symmetric spaces, which are particular homogeneous spaces. We classify simple Lie algebra over C according to Dykin diagram. Also by means of real form we classify simple Lie algebra over IR. We show that every symmetric space gives rise to an orthogonal symmetric Lie algebra. Finally we classify Riemannian symmetric spaces of types I, II, III and IV according to the classification of the irreducible orthogonal symmetric Lie algebra of types I, II, III, and IV.enSymmetric SpacesThesis