Suppose M and N are smooth manifold with M connected, and F:M->N is a smooth map and its pushforward is zero map for each p in M. Show that F is a constant map. I just remember from topology, the only continuous functions from connected space to {0,1} are constant functions. With this be useful in solving the problem?
Show that if M is connected, then a locally constant map F:M-->N is (every pt has an open nbhd where F is constant) is constant. Then show that a vanishing pushfoward implies that F is locally constant.
If you define a path between arbitrary points of M, your domain becomes [0,1] which is easier to work with. If you're using Lee (I think I remember this exercise) you have as a theorem that connected manifolds are path connected.