Timeline for What is soliton
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 3, 2018 at 9:44 | comment | added | Qfwfq | Thank you. In reading your (very apt) example I was confused at first, because $g_0$ and $\phi_t^*(g_0)$ are not conformally equivalent, in general. Still, diffeomorphisms preserve intrinsic properties of metrics (such as being conformally flat, indeed). | |
| Sep 3, 2018 at 9:26 | comment | added | Ben McKay | @Qfwfq: take any metric. Its exponential map identifies a ball with a ball in its tangent space (radius=injectivity radius). The family $g_t=(1-t)g_0+tg_1$ of metrics in that ball starts at the original metric $g_0$ and ends at the induced Euclidean metric $g_1$ on the tangent space. In some smaller ball, this is a metric for all $t$. If the original metric is not conformally flat, for example, taking the Fubini-Study metric on the complex projective plane, then you have a family which contains a confomally non-flat and a flat, so is not a diffeomorphism followed by a conformal rescaling. | |
| Sep 2, 2018 at 22:22 | comment | added | Qfwfq | Could you [or anybody else.. I realize this answer is from 2012] make an elementary example of a family $g(t)$ of metrics which is not of the form $\alpha(t)\cdot \phi_t^*(g(0))$ for $\alpha(t)$ a (time dependent) scalar and $\phi_t$ a family of diffeomorphisms? | |
| Mar 6, 2012 at 2:12 | vote | accept | zapkm | ||
| Mar 5, 2012 at 19:05 | history | answered | Michael Coffey | CC BY-SA 3.0 |