Embedding of closed manifolds of same dimension

Question

I have a an embedding $F: M \longrightarrow N$ of two closed smooth manifolds $M, N$ of the same dimension. Do I have a diffeomorphism automatically?

Answer 1

The image of $F$ is closed (as $F$ is continuous and $M$ compact) and open (as $F$ is an embedding) in $N$, so if $N$ is connected then $F(M)=N$ and you indeed get a diffeomorphism.