How does one prove that local diffeomorphism is submersion?

Question

1. How does one prove that local diffeomorphism is submersion?
2. For a manifold, what does it being disconnected mean? I get what "disconnected" means for a graph, but not for a manifold.

Answer 1

1. If $f:M\to N$ is a local diffeomorphism, then for any point $p\in M$, there's a neighborhood $U$ of $p$ in $M$ such that $f|_U: U\to f(U)$ is a diffeomorphism. Show that $$\mathrm{d}(f|_U)_p:T_pM\to T_{f(p)}N,\qquad \mathrm{d}f_p:T_pM\to T_{f(p)}N$$ are the same map, and then use that an isomorphism of vector spaces is necessarily surjective.
2. It means that, as a topological space, it is not connected. See the relevant Wikipedia article.