So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle $TU$ is diffeomorphic to $M\times S^1$.
So, I can see that at each point of $M$ the unit tangent space is isomorphic to $S^1$, but I don't see how to use the nowhere vanishing vector field.
First, I need to convince myself that over small enough neighborhoods we have a local trivialization. Then maybe by compactness I could cover the manifold with finitely many trivially covered neighborhoods. And maybe then use the nowhere vanishing vector field to connect them somehow, but I am not sure.
Or, maybe there is a clever way of writing the diffeomorphism using the vector field.
Any hints or thoughts would be appreciated, thanks in advance!