The set of Morse point is open and dense, Exercise 3.3.23 in DoCarmo's Differential Geometry

Question

Let $S\subseteq\mathbb{R}^3$ be a smooth surface, no boundary (not necessarily compact). For any $r\in\mathbb{R}^3\backslash S$, define a smooth function $h_r:S\to\mathbb{R},q\mapsto |q-r|$

Let $p\in S$ be a critical point of $h_r$, then $\vec{pr}\perp T_p S$. So $p$ is non-degenerated if $\frac1{|p-r|}\neq k_1,k_2$. $k_1,k_2$ are two principal curvatures ($\vec{pr}$ be the normal direction at $p$, $k_{1,2}$ need not to be different).

Call $h_r$ a Morse function, if every critical point $p\in S$ of $h_r$ is non-degenerated.

Show that $\{r\in\mathbb{R}^3\backslash S|h_r$ is Morse funtion$\}$ is a open and dense set in $\mathbb{R}^3$.

I find little useful information of the set, possibly its complement is not a discrete set.

Answer 1

HINT: Consider the sets $\{x\in\Bbb R^3: |x-p| = 1/k_1(p) \text{ for some } p\in S\}$ and $\{x\in\Bbb R^3: |x-p| = 1/k_2(p) \text{ for some } p\in S\}$.