Parallel vector spaces on Riemannian manifolds

In this thesis, after the definition of differentiable manifold, the tangent vector notion is defined via differentiable curves. It is given some basic notions such as cotangent vector, vector field, 1-form and tensor field. The fundamental theorem of Riemannian manifolds is differently reestablished and then it is proved. Riemannian curvature, Ricci curvature and scalar curvature are defined. Some differential operators are given in Riemannian manifolds Two theorems including parallelism of Killing vector fields and Harmonic vector fields are proved.