Is there any example where $\nabla F=0$ and the level surface or level curve is smooth and has a normal?

Question

Consider a function $F:\mathbb{R}^n\to\mathbb{R}$ ($n$=2 or 3). Is there any example where $\nabla F(a)=0$ but the level surface or level curve has a non vanishing normal?

Answer 1

Take $f(x,y)=(y-x)^2$ then $\{f=0\}$ is the line $y=x$ which has a tangent and a normal but $\nabla f(x,x)=(0,0)$.

Answer 2

How about $$F(x,y) = x^3?$$

$\nabla F(0,y) = 0$ but the level set $F^{-1}(0)$ has a "normal" $(1,0)$.

In general we try to avoid these cases. As you may be aware, if $a$ is a regular value of $F$, then the inverse image $F^{-1}(a)$ is guaranteed to be a regular submanifold of $\mathbb{R}^n$, which means it has a constant dimension without singularities, degenerate points, etc and has well-defined normal $\nabla F / \|\nabla F\|$ everywhere. Moreover the condition of being a regular value is not so onerous since by Sard's lemma this is true for almost all $a$.