Timeline for Surjectivity of the normal exponential map
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 22, 2013 at 17:44 | answer | added | Peter Michor | timeline score: 2 | |
| Jan 25, 2010 at 21:13 | comment | added | Will Jagy | I think a stricter condition you may want to try for noncompact preimage N is that the embedding or immersion is a proper map, for Riemannian manifolds sometimes abbreviated "proper embedding" or "proper immersion." See en.wikipedia.org/wiki/Proper_map and surprising projecteuclid.org/… | |
| Jan 25, 2010 at 14:33 | comment | added | CuriousUser | actually, embedding is not enough, since the spiral approaching a circle is, in fact, embedded in the plane (the induced topology is the topology of the real line). I think that the "clear" condition, is $f$ closed. | |
| Jan 25, 2010 at 14:26 | comment | added | CuriousUser | (1-2) Thank you, i was actually thinking about a spiral approaching a circle, as you said. (3) First, i would like to know if there are papers about this topic. Secondly, i look at cases where $N$ doesn't "get closer and closer to itself", as in the spiral case. In other words, $N$ is a leaf of a Riemannian foliation, defined on an open set $V\supset N$. In particular, for any $x\in \overline{N}\cap V$, there is a neighbourhood $U$ such that the connected components of $N\cap U$ are costant distance apart. But since this riemannian foliation is not defined everywhere, it's hard to be precise.. | |
| Jan 25, 2010 at 5:48 | comment | added | Anton Petrunin | (1) in your example with logarithmic spiral $N$ is not complete, but you can build a spiral to make it complete. (2) It seems not sufficient to ask that you have a bound on second fundamental form; a similar spiral-like example can be build which approach a circle. (3) embedding is clearly enough, but it is not what you want. --- BUT what do you want...? | |
| Jan 25, 2010 at 4:11 | history | asked | CuriousUser | CC BY-SA 2.5 |