الفهرس 
Introduction and definitions of RiemannianOnly 14 pages are availabe for public view
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Abstract
This chapter presented the definitions of smooth manifolds, submanifolds and Riemannian manifolds and presented many of the geometric objects defined on manifolds such as scalar fields, tangent vectors, tangent space, vector fields
and tensor fields. This chapter discussed parallel vector fields of constant length, arc length, geodesics, connections and Riemannian connections.
Also it presented a detailed study for: Cartan structure equations, curvature tensor, Ricci tensor and sectional curvature on the Riemannian manifold. This chapter studied vector fields on Riemannian manifolds. Some new additional conditions are introduced for the vector fields to be finite killing on Riemannian manifolds. Also we defined an integral formula for killing vector fields which help in proving theorem (3.4.1).