Vector bundles on manifolds: the cohomology of projective algebraic varieties

This Thesis looks at different areas of mathematics which require the notion of a differentiable manifold. The first three chapters contain various materials such as differentiable manifolds, submanifolds, tangent bundles, differential forms, integrations, Stokes' theorem, and certain concepts of Riemannian geometry. In the last chapter, we show that the kernel of a given elliptic operator L is finite dimensional. We also show that on compact differentiable manifolds for each de Rham cohomology class a there exists a unique harmonic form y, showing that the de Rham cohomology is finite dimensional.