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First Let me state what I know,

If $r$ is a regular value of $F$ and $F^{-1}(r)$ is non-empty then $F^{-1}(r)$ is a regular surface

Of course I know, its converse is not true. Typical example can be found in Do Carmo as $F(x,y,z) = z^2$. The critical points of this surface is $(x,y,0)$ so it has a critical value $F(x,y,0) =0$. But $F^{-1}(0) = \{(x,y,0)\}$ is a plane so it is a regular surface.

Now I am trying to solve following problem

For which real numbers $c$ is the set of $(x,y,z)$ satisfying $x^2+y^2 + \sin(z) =c$ a regular surface?

Applying the above procedure. Let $F(x,y,z) = x^2 + y^2 + \sin(z)$, then $\nabla F = (2x,2y, \cos(z))$ so $(0,0, \frac{\pi}{2}+n\pi)$ is a critical point and thus critical value is $\pm 1$. So $c\neq \pm 1$. $F^{-1}(c)$ is a regular surface.

Now I want to know what happen to $c=\pm 1$. For this case $F^{-1}(\pm 1) =\{ (x,y,z) | x^2 + y^2 + \sin(z) =\pm 1 \}$. At this moment I am stuck with this stage.... For this do i have to compute $X(u,v) = ( \pm \sqrt{-u^2 - \sin(v)}, u, v)$ and compute $X_u \times X_v$?

Is there any systematic way to find regularity of surface for critical value? i.e., I want to find some general strategy to solve Find real numbers $c$ satisfying $F(x,y,z) = c$ be a regular surface.

phy_math
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  • No, $\pm 1$ are critical values. If you think about Taylor series, you'll see that the surface looks like the cone $x^2+y^2=w^2$ near the critical points. – Ted Shifrin Sep 12 '19 at 21:33
  • @TedShifrin I didn't quite see what you mean by using Taylor series. Could you please elaborate? Thanks. – Tengu Sep 13 '19 at 01:48
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    @Tengu: At $(0,0,\pi/2)$, say, what is the second-degree Taylor polynomial of $F$, writing $z-\pi/2 = w$? – Ted Shifrin Sep 13 '19 at 02:18
  • @TedShifrin Oh, I see, it's $x^2+y^2=w^2/2$. – Tengu Sep 13 '19 at 02:40

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I would say that the following proposition may be of use:

For a point $p$ in a regular surface $S\subset \mathbb R^3$, there exists a neighborhood $V$ of $p$ in $S$ such that $V$ is a graph of a differentiable function which has one of the following three forms: $z=f(x,y),y=g(x,z), x=h(y,z)$.

Now, from $x^2+y^2+\sin z = 1$, you can write one variable as function of the other two and the rest is to show that at such function is not differentiable at some point ($(0,0,\pi/2)$ for example).

Tengu
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