I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map.
I know that there exists a one-parameter group of diffeomorphisms that are smoothly homotopic to the identity, in fact the global flow of this vector field plays this role, since $\theta:\Bbb{R}\times M\to M$ is a smooth left-action.
My question is: How can I show that exists $t_0\in\Bbb{R}$ such that $\theta_{t_0}$ has no fixed points? (Here $\theta_{t_0}(x)=\theta(t_0,x)$.
Thanks!