Timeline for Global invertibility
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 4, 2011 at 21:10 | comment | added | algori | ... this is a local diffeomorphism with image $\mathbb{C}=\mathbb{R}^2$ but not a global diffeomorphism as it takes each value infinitely many times. | |
| Dec 4, 2011 at 21:08 | comment | added | algori | stephanos -- re "it is a global diffeomorphism ...": it isn't in general: take $f$ to be the complex exponential, then $M=\mathbb{R}^2$ minus the origin, which is not diffeomorphic to the plane. Even if the image of $f$ is the whole $\mathbb{R}^n$, then $f$ is not necessarily a diffeomorphism. To see this set $g$ to be a degree 3 complex 1-variable polynomial whose derivative has no double roots. The universal cover of $U=\mathbb{C}\setminus$ the roots of $f'$ is $\mathbb{R}^2$; set $h$ to be the universal covering map $\mathbb{R}^2\to U$. Set $f=g\circ h$.... | |
| Dec 4, 2011 at 20:41 | vote | accept | Sz_Z | ||
| Dec 4, 2011 at 18:20 | history | answered | stephanos | CC BY-SA 3.0 |