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The main aim of this thesis is to study the theory of Sobolev spaces on Riemannian manifolds. This thesis is divided into three parts, 1st we will learn Riemannian Geometry then Sobolev space on R n at last we will define Sobolev space on Riemannian Manifolds and we will learn some properties and embeddings of Sobolev space on Riemannian Manifolds. The Sobolev space over R n is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces. The functions of Sobolev space is not easy to handle, we shall approximate this functions by smooth functions. We have calculated some inequalities on Sobolev space. With the help of this inequalities we will embedded the Sobolev space in some L p space and H¨older continuous space. Similarly on the manifold using covarient derivative we define Sobolev Space Over Riemannian Manifold. Riemannian manifolds are natural extensions of Euclidean space, the naive idea that what is valid for Euclidean space must be valid for manifolds is false. But Sobolev embedding theorem for R n does hold for compact manifolds.

The main aim of this thesis is to study the theory of Sobolev spaces on Riemannian manifolds. This thesis is divided into three parts, 1st we will learn Riemannian Geometry then Sobolev space on R n at last we will define Sobolev space on Riemannian Manifolds and we will learn some properties and embeddings of Sobolev space on Riemannian Manifolds. The Sobolev space over R n is a vector space of functions that have weak derivatives. Motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces. The functions of Sobolev space is not easy to handle, we shall approximate this functions by smooth functions. We have calculated some inequalities on Sobolev space. With the help of this inequalities we will embedded the Sobolev space in some L p space and H¨older continuous space. Similarly on the manifold using covarient derivative we define Sobolev Space Over Riemannian Manifold. Riemannian manifolds are natural extensions of Euclidean space, the naive idea that what is valid for Euclidean space must be valid for manifolds is false. But Sobolev embedding theorem for R n does hold for compact manifolds.