$M^n\subset \mathbb{R}^{n+1}$ isometric immersion, $R\equiv 0\Rightarrow M$ locally isometric to $\mathbb{R}^n$?

Question

Let $f:M^n\to\mathbb{R}^{n+1}$ be an isometric immersion with $R\equiv 0$, where $R$ is the curvature tensor of $M$.

Can we say that $M$ is locally isometric to $\mathbb{R}^n$?

I know that by Killing-Hopf theorem, we can say that $M$ is globaly isometric to $\mathbb{R}^n$, provided that $M$ is complete and simply connected.

If I consider an open subset $U\subset M$, I can assume it to be simply connected, but not complete, so I can't apply Killing-Hopf.

I have the feeling that the answer is yes, but I don't know how to prove it.

Any suggestions?