Timeline for Surface homeomorphic to an open disc with two congruent curves homoemorphic to a circle
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 29, 2011 at 6:11 | comment | added | Ehsan M. Kermani | @Robert Bryant: Is there any chance to find a sufficient and necessary conditions in that case? | |
| Dec 25, 2011 at 2:37 | comment | added | Ehsan M. Kermani | @Robert Bryant: Thank you. I really appreciate it! | |
| Dec 24, 2011 at 2:06 | comment | added | Robert Bryant | @ehsanmo: Take a surface $S$ in Euclidean $3$-space that has constant Gauss curvature $-1$. Then any two closed curves on $S$ with sufficiently large constant geodesic curvature $\kappa >> 0$ will close at the same length and will be congruent in the first sense, but because the intrinsic isometry that identifies them will not be an ambient isometry, the two curves won't usually be congruent in the second sense. OTOH, there is a surface with two closed curves that are ambiently congruent but don't have the same geodesic curvature in $S$, so they are not congruent in the first sense. | |
| Dec 23, 2011 at 19:18 | comment | added | Ehsan M. Kermani | @Robert Bryan: Could you enlighten me the difference? | |
| Dec 23, 2011 at 12:50 | comment | added | Robert Bryant | @Gjergji: Thanks for the reference. @ehsanmo: In fact, the solution described in the passage Gjergji supplied assumes the latter definition of 'congruent', rather than the former. It is a completely different problem. | |
| Dec 23, 2011 at 6:36 | comment | added | Gjergji Zaimi | Here is the solution to the MS problem books.google.com/… | |
| Dec 23, 2011 at 1:54 | comment | added | Ehsan M. Kermani | I'm assuming the former definition. | |
| Dec 23, 2011 at 1:26 | comment | added | Robert Bryant | I don't understand the problem because I don't know what definition of 'congruent' you mean. Are you assuming that the surface has a (Riemannian) metric on it (and 'congruent' means isometric with corresponding geodesic curvatures) or that it is embedded in $3$-space (and 'congruent' means extrinsically congruent as curves in $3$-space)? | |
| Dec 23, 2011 at 0:22 | history | asked | Ehsan M. Kermani | CC BY-SA 3.0 |