Can anyone provide a diffeomorphism between these "spheres": $\mathbb{S}^2$ and $\{(x,y,z)\in \mathbb{R}^3: x^4+y^2+z^2=1\}$?
PS: If you know a result that can solve this problem, I would be glad to know.
Can anyone provide a diffeomorphism between these "spheres": $\mathbb{S}^2$ and $\{(x,y,z)\in \mathbb{R}^3: x^4+y^2+z^2=1\}$?
PS: If you know a result that can solve this problem, I would be glad to know.
The problem is $sign(x)\sqrt{x}$ and $sign(x)x^2$ are not strictly speaking diffeomorphisms from $[-1,1]$ to $[-1,1]$ since the differential at $x=0$ either does not exist or vanish.
This is fixable by partition of unity. Take the suggested map $g: (x,y,z)\mapsto(sign(x)\sqrt{x},y,z)$ on the domain where $|x|>1/4$.
And on $|x|<1/2$ the projection map $h: (x,r,\theta)\mapsto(x,r'=\sqrt{x^2-x^4+r^2},\theta)$ (here the obvious $(y,z)$ into polar coordinates $(r,\theta)$ is used.)
These two maps can be patched together using a partition of unity $f=\lambda g + (1-\lambda) h$ where $\lambda$ is a smooth function $\lambda:\mathbb{R}^3\rightarrow \mathbb{R}$ such that $\lambda=1$ for $|x|\ge 1/2$, $\lambda=0$ for $|x|\le 1/4$.