I am teaching a summer qualifying exam class, and a student was looking up practice questions online. I cannot figure one of them out.


I am trying to solve this problem: Let $M$ be an $n$-dimensional manifold embedded in $\mathbb{R}^{n + 1}$. Then almost every hyperplane in $\mathbb{R}^{n + 1}$ is not tangent to $M$ at any point.


The hint given is to consider the map $f: M \to S^n$ that takes $x \in M$ to the unit normal at $x$.


I first thought to use Sard's Theorem and analyze the critical values of this map. Then I found examples where there are points with tangent hyperplanes, but $f$ has no critical values. I've tried to define other maps and analyze them, but can't produce a map that has critical values precisely where I want them.


I'd like if the solution was via the hint, but any solution is welcome.



Almost All Hyperplanes are Not Tangent
This should be a little easier than using Sard's theorem:


The manifold $H$ of hyperplanes in $(n+1)$-space has dimension $n+1$. The map from $M$ to $H$ that sends each point to its tangent hyperplane is a differentiable map from an $n$-dimensional manifold to an $(n+1)$-dimensional manifold. The image must have measure zero. 


You just need a lemma saying that if $f: U \rightarrow \mathbb{R}^m$ is a $C^1$ map from an open subset of $\mathbb{R}^k$ to $\mathbb{R}^m$ with $k < m$, then the image has measure zero. This is also a lemma for the Morse-Sard theorem (see Hirsch's Differential Topology). 


