The Analytical techniques on manifold via differential forms

Abstract

In this research, we show by using integration of different form that , for a compact orientable manifold of dimension n the De Rham cohomology group 􀜪􀯡(􀜯) is non zero, we also show that the group for a compact , connected , orientable manifold is just one-dimensional. We discuss the metric tensor and Riemannian metric on a manifold in informal terms. We also illustrate the relation between the integral curve of geodesic flow and geodesic with some examples and applications .