problem of Math 8302 , Manifolds and Topology II posted

According to the assignment, M may not be path-connected. Thus we decompose M into path-connected components. Since M is locally Euclidean, the path-connected components are the same as the connected components and are open subspaces of M . Thus each component itself is a smooth manifold. Then we will solve the problem on each of them separately but with the same method. This means we can assume from the beginning that M is connected. Our claim is that ω satisfies the conditions stated in the problem if and only if ω(t) is independent of t and ω is an exact 1-form on M . In other words, there is a smooth map η : M → R such that ω = dη. We will break our proof into 3 steps, namely M = R, M = R and M arbitrary.