$k$-dimensional manifolds in $\mathbb{R}^m$ with $k<m$ are non-dense and have measure zero

Question

Let $V \subset \mathbb{R}^m$ be a smooth (connected) $k$-dimensional manifold with $k<m$. Is it then true that $V$ is non-dense and has measure zero?

It seems a pretty straightforward question. I had no success in finding a suitable theorem/lemma.

Answer 1

Hint: Just look at Sard's Theorem and take $f = \iota$ which is the inclusion.