Timeline for Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric
Current License: CC BY-SA 4.0
18 events
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| Sep 5, 2020 at 11:17 | vote | accept | CommunityBot | ||
| Sep 5, 2020 at 11:15 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 5, 2020 at 9:18 | history | edited | user164740 |
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| Sep 5, 2020 at 3:32 | history | edited | C.F.G |
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| Sep 5, 2020 at 0:21 | answer | added | Robert Bryant | timeline score: 15 | |
| Sep 3, 2020 at 22:13 | comment | added | Qfwfq | (I edited a bit incorporating the remark of abx in the OP's question) | |
| Sep 3, 2020 at 22:12 | history | edited | Qfwfq | CC BY-SA 4.0 |
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| Sep 3, 2020 at 21:53 | comment | added | Jason DeVito - on hiatus | @Qfwfq: If rotations by $\pi/2$ at every point are isometries, then so are rotations by $\pi$. Thus, the metric on $S^2$ exhibits $S^2$ as a symmetric space. This implies the metric on $S^2$ is isometric (up to scaling) to the usual round metric. So, no, there is no such metric distinct from the $FS$ metric. | |
| Sep 3, 2020 at 21:46 | history | edited | user164740 |
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| Sep 3, 2020 at 21:18 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 3, 2020 at 21:07 | comment | added | Qfwfq | @alglearner: the metric on the sphere would have to be invariant under rotations of $\pi/2$ at every point (see comments to my wrong answer). Is there such a metric on the $2$-sphere, distinct from FS? | |
| Sep 3, 2020 at 21:03 | comment | added | aglearner | Take $\mathbb CP^1$ with Fubini-Studi metric of area $1$, and take any other Kaher metric on $\mathbb CP^1$ which has area $1$ but has non-constant curvature. Then $\psi$ doesn't exist, because the manfiolds are not isometric and your condition implies that $\psi$ is an isomery. But they are both symplectomorphic and obviously biholomorphic. | |
| Sep 3, 2020 at 21:00 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 3, 2020 at 20:04 | history | edited | user164740 | CC BY-SA 4.0 |
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| Sep 3, 2020 at 19:56 | comment | added | abx | On a Hermitian manifold, the symplectic and the complex structure determine the metric. | |
| Sep 3, 2020 at 19:28 | comment | added | LeechLattice | @Qfwfq It probably means an isometry that preserves the complex and the symplectic structure at the same time. | |
| Sep 3, 2020 at 19:27 | comment | added | Qfwfq | What do you mean by "isomorphic" Kaehler manifolds? Is isometry is implied in your definition of isomorphism? | |
| Sep 3, 2020 at 19:05 | history | asked | user164740 | CC BY-SA 4.0 |