Compact hypersurface

Question

I want to show that for some compact hypersurface $M$ in $\mathbb{R}$ it holds that for any $v \in S^n$ there exists some $p \in M$, s.t. $v$ is the unit normal to $T_pM$. I also have the hint to look at the hyperplanes $Rv+v^{\perp}$ for some $R>0$, s.t. this hyperplane and $M$ are disjoint and then to look at the point with minimal distance to $Rv+v^{\perp}$.

Now I can picture this (for $n \leq 3$ obviously) because then $Rv+v^{\perp}$ is some plane $\cong \mathbb{R}^2$ and if I take the point with minimal distance its obvious (graphically) that its tangent space has to be orthogonal to $v$. But I don't know how to formalize this.

I can get the point with minimal distance to $Rv+v^{\perp}$ by looking at the smooth function $f:M \rightarrow \mathbb{R}, q \mapsto dist(q,Rv+v^{\perp})$ which has a minimum since $M$ is compact.

Can somebody give me some hint how to formalize this (for all dimensions)? Thanks!

Answer 1

Let $v$ be a unit vector. Consider the map $f: M \rightarrow \mathbb{R}$ such that $f(x)=x \cdot v$.

Then, at any $x \in M$, $d_x f: X \in T_xM \subset \mathbb{R}^n \longmapsto v \cdot X$. There exists some $x \in M$ at which $f$ is maximal.

Then $d_xf=0$, thus $T_xM \subset v^{\perp}$, hence $T_xM=v^{\perp}$.

Answer 2

Here is a variational approach which (for $N$ a proposed hyperplane) follows the hint:

Assume $M$, $N$ are submanifolds of $\mathbb R^n$ and there are points $p\in M$, $q\in N$ such that $d(p,q)=\inf_{m\in M,n\in N} d(m,n)>0$. Set $v=p-q$. Then $v\perp T_pM$ and $v\perp T_qN$.

$\textbf{proof:}$

Let $w\in T_pM$ and choose a curve $\gamma: (\varepsilon,\varepsilon)\to M$ with $\gamma(0)=p$, $\gamma'(0)=w$ and define $\delta (t)=d(\gamma(t),q)^2$. Then as $\delta$ achieves a minimum at $0$:

$$0=\delta'(0)=\langle \gamma(t)-q,\gamma(t)-q\rangle'(0) =2\cdot \langle\gamma'(0),\gamma(0)-q\rangle=2\langle w,v\rangle$$ Hence $v\perp T_pM$ and then by symmetry also $v\perp T_qN$.