Problem: Let $M \subset \mathbb{R}^{n+1}$ be a compact submanifold ($\dim M=k$) and $n \geq 2k + 1$.
Show that, for the projection $\pi : \mathbb{R}^{n+1} \longrightarrow H^n$ onto a suitable hyperplane of $\mathbb{R}^{n+1}$, the restriction $\pi|M : M \longrightarrow H$ is a smooth embedding.
I tried to use a corollary from Whitney's Embedding Theorem that says that if $M$ ($\dim M = k$) is a compact manifold with or without boundary and $n \geq 2k + 1$, then every smooth map from $M$ to $\mathbb{R}^n$ can be uniformly approximated by embeddings.
The problem is I don't actually see how this could work here, so any hints and maybe some intuition on what is happening would be really appreciated.
Thanks.
A series of hints, which should lead you to the solution
1: Try to show that there exists such a hyperplane, by showing that the set of all such hyperplanes has full measure.
2: Note that a hyperplane corresponds to a line and vice versa. We have found a bijection by taking the orthogonal complement!
3: The projection injects iff the corresponding line based at any point hits the embedded manifold at most once.
4: Lines are described by the space $RP^n$. Is this a smooth manifold? If yes which dimension?
5: Try to find all elements of $RP^n$ which correspond to a non-injective map. Try to do so as the image of a smooth map.
6: Try to calculate dimensions to use Sard to show that the image of the above map is a measure zero set.
Finally note that you might want to try to use the hints not in a particular order.