Let $M \subseteq \mathbb{R}^n$ be a m-dimensional manifold. Suppose $n>2m+1$. Show that there is a projection from $M$ to a (2m+1)-dimensional subspace of $\mathbb{R}^n$ so that the image is a manifold in $\mathbb{R}^{2m+1}$ by following the steps below
i) Show that is suffices, for $n > 2m+1$ to find a non-zero vector $a \in \mathbb{R}^{n}$ and a projection $\pi_a: \mathbb{R}^n \to a^{\bot}$, where $a^{\bot}$ is a codimension one subspace perpendicular to $a$ so that $\pi_a(M)$ is a manifold in $a^{\bot}$.
ii) Define $g: M \times M \times \mathbb{R} \to \mathbb{R}^n$ and $h: TM \to \mathbb{R}^n$ by: $$g(x,y,t) = t(x-y)$$ $$h((p,v)) = v,$$ where $(p,v)$ represents the vector $v \in T_p M$. Show that $\pi_a(M)$ is a manifold in $a^{\bot}$ if $a$ is not in the images of $g$ and $h$.
iii) Apply Sard's Theorem.
This is an exercise that we have to do for our course on bifurcation theory. Here is all we know by now:
For the first part we think we have to use induction, to keep reducing the dimension.
However, for the second part we are at a loss. We somehow see why, if $a$ is either in the image of $g$ or $h$, that then $M$ can't be a manifold, because you'd get self intersections. The thing is that this doesn't prove what is asked.
Also, for part iii), we don't see how to apply Sard's theorem, because what is says is:
Let $f: M \to N$ be a smooth map between manifold $M,N$. Then the set of regular values is dense.
Thanks in advance for any help.
Suppose $a$ is in the second image (image of $h$). Then there's a point $p$ whose tangent space contains the vector $a$. Projection on to $a^\perp$ will send this tangent vector to $0$, so the projection isn't maximal rank on every tangent space, hence it's not an embedding (or even an immersion!).
What about if $a$ is in the image of $g$? That'd mean that there are two distinct points $x$ and $y$ in $M$ such that $P(x) = P(y)$, where $P$ is projection along $a$, so again it's not an embedding.
For part iii, I leave you to your own devices for now. The trick is to figure out "what function do I apply the theorem to?" Hint: it's not either $g$ or $h$. What kind of value do you want to get OUT of Sard? (Answer: a projection direction $a$). So what should be the domain of your map $f$?