Projection of a Manifold has measure zero

Question

Let $M$ be a $k$-dimensional sub manifold of $\mathbb{R}^N$, and let $\pi_n:\mathbb{R}^N\to\mathbb{R}^n$ be the canonical projection, with $n>k$. Can we show that $\pi_n(M)\subset \mathbb{R}^n$ has measure zero?

Help much appreciated! :)

Answer 1

Look up Sard's theorem. This answers this question in more generality, and I doubt that it is too much easier without this generality. A condensed version is

If $f:\Bbb R^k\rightarrow \Bbb R^n$ is a $C^{\infty}$ function, the image of the set of points where the total derivative is not surjective is measure zero in $\Bbb R^n$ (note no assumptions on $n$ or $k$, but if you fix $n>k$ then the total derivative is not surjective anywhere).

To apply this to your problem, you are going to want to choose a countable chart for your manifold and use $\sigma$-additivity.