If $\operatorname{dim} M > \operatorname{dim} N$, is there an injective smooth map $M\to N$?

Question

Let $m>n$ and suppose $M$ is a smooth $m$-manifold, $N$ is a smooth $n$-manifold.

Can there be an injective smooth map $f:M\to N$?

Answer 1

Hint: Considering the problem locally, this requires an injective smooth map $\mathbb R^m\to \mathbb R^n$.

Answer 2

The answer is no, and in fact there cannot even be an injective continuous map $M\to N$.

As noted by Hagen von Eitzen, we look at the map locally, so that it suffices to show that there is no continuous injection $\mathbb{R}^m\to\mathbb{R}^n$. This can be done using the Borsuk-Ulam theorem, as done by Jason DeVito in his answer to Why isn't there a continuously differentiable injection into a lower dimensional space?.