I cannot prove any conclusion of this problem. Can anyone please help me? Let $f:M\to N$ be an immersion of $M$ into $N$ and dim $M=\dim N$.Prove or disprove that $f(M)$ is a submanifold. Thanks for any help.
$f(M)$ is an open subset of $N$ and thus a submanifold. Proof by unwrapping of definitions: say $y=f(x)\in f(M)$. Since $f$ is an immersion $df(x)$ is injective, and since $\dim(M)=\dim(N)$ $df(x)$ is in fact an isomorphism. By the inverse function theorem $f$ restricted to some neighborhood of $x$ is a diffeomorphism onto a neighborhood of $y$. In particular, $f(M)$ contains a neighborhood of $y$.
Use this. An immersion is an open map if (and only if, I think) $\dim M = \dim N$. This is because for $\dim M = \dim N$, we have that immersion, submersion are local diffeomorphism are equivalent.
Images of open maps are open subsets. (Conversely, image open subset implies open map, I think.)
Open subsets can always be made into submanifolds of codimension zero. (Conversely, underlying sets of submanifolds of codimension zero are open subsets, I think.)