$f:M \to N$ is a diffeomorphism iff $J(f)\ne 0$?

Let $M$ and $N$ be two smooth manifolds and $f:M\to N$ a smooth map. By definition:

$f$ diffeomorphism $\Leftrightarrow$ $f$ is invertible and $f^{-1}:N\to M$ is smooth.

It is clear that if $f$ is a diffeomorphism then the induced map $f_{\star p}:T_pM\to T_{f(p)}N$ is an isomorphism. Thus:

$f$ diffeomorphism $\Rightarrow$ the Jacobian matrix $J(f)$ is regular in any local chart of $M$

Is the converse true?:

$$\det J(f)(p)\ne 0 \,\text{for any}\, p\in M\,\text{implies that}\, f\,\text{is a diffeomorphism}?$$

Solution 1:

No. Take $M = \mathbb R$, $N = S^1$ and $f(t) = (\cos t, \sin t)$.