TITLE: Is this surface diffeomorphic to a 2-sphere?
QUESTION [18 upvotes]: Let  $f:\mathbb{R}^3\to \mathbb{R}$ be defined by $f(x,y,z)=x^4+y^6+z^8$. Let $M=f^{−1}(1)$.
Is $M$ is diffeomorphic to a sphere $S^2$?

I tried to solve this problem, but I realized that I have no tools to solve it.
The constant rank theorem tells me $M$ is a smooth 2-dimensional manifold, but does not tell me how it looks like.
And more generally, when is $N=\{x,y,z\in\mathbb{R}^3\mid ax^n+by^m+cz^l=1\}$ diffeomorphic to a sphere? What tools can I use to solve this problem?
Thank you for reading. Hoping get some shedding light in your reply.

REPLY [5 votes]: Morse Theory is a very suitable tool to solve this kind of problems. We shall find a Morse function on $M$ and we can compute the indices of its critical points. Take
$$
\begin{array}{rccc}
h:&M&\longrightarrow&\mathbb{R}\\
  &(x,y,z)&\longmapsto & x
\end{array}.
$$
Note that $T_{(x,y,z)}M=\ker d_{(x,y,z)}f$. Since
$$
df=4x^3dx+6y^5dy+8z^7dz
$$ 
its kernel is generated by
$$
3y^5\frac{\partial}{\partial x}-2x^3\frac{\partial}{\partial y}\ \mbox{and}\ 2z^7\frac{\partial}{\partial x}-x^3\frac{\partial}{\partial z},\ \mbox{at points with}\ x\neq 0
$$
$$
\frac{\partial}{\partial x}\ \mbox{and}\ 4z^7\frac{\partial}{\partial y}-3y^5\frac{\partial}{\partial z},\ \mbox{at points with}\ x=0
$$
Now, we have $dh=dx$ which applied to the vector fields above gives the functions $3y^5,2z^7$ at points with $x\neq 0$ and the constants $1,0$ at points with $x=0$. From this it follows that $(x,y,z)\in M$ is a critical point of $h$ if and only if $y=z=0$. So $h$ has exactly two critical points, which are $(1,0,0)$ and $(-1,0,0)$.
Finally, since $M$ is compact --I assume you know how to prove this-- and admits a Morse function with only two critical points, it must be a sphere (in this very simple case there's no need to compute the indices of the critical points, although they are 0 and 2, of course).
As for the general case $ax^n+by^m+cz^l=1$ you can carry out a similar computation and decide what surface you get in terms of critical points and its indices. However you'll need to be careful with compactness (what happens if one of the exponents is odd?) and probably you'd like to assume $abc\neq 0$.