MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

let $f:R^3 -> R \ \ \ \ \ \ \ \ f(x,y,z)=x^4 + y^6 +z^8 \\$

$M = f^{-1}(1)$

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.

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 $N = \{ x,y,z \in R^3 | ax^n + by^m + cz^l = 1\}$ is 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.

share|cite|improve this question

closed as off topic by Chris Gerig, Benoît Kloeckner, Mariano Suárez-Alvarez, Martin Brandenburg, Willie Wong Feb 21 '13 at 12:09

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

1 is a better place for these questions. – Mariano Suárez-Alvarez Feb 21 '13 at 8:51

It is. The homeomorphism from $X$ to the unit sphere is $$ x\mapsto \mathrm{sign}(x)\cdot x^2,\ y\mapsto y^3,\ z\mapsto\mathrm{sign}(z)\cdot z^4. $$

share|cite|improve this answer

Here is a plot of the surface $x^4+y^6+z^8 = 1$. It is approximately a cube with opposite corners $(-1,-1,-1)$ and $(1,1,1)$.
           Implicit Surface

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.