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## Main Question or Discussion Point

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?

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?