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.
