Timeline for What are some explicit examples of nontrivial gradient almost Ricci solitons with harmonic curvature?
Current License: CC BY-SA 4.0
15 events
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| Dec 2, 2022 at 15:33 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo
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| Nov 28, 2022 at 15:45 | vote | accept | Matheus Andrade | ||
| Nov 28, 2022 at 11:56 | answer | added | Robert Bryant | timeline score: 4 | |
| Nov 27, 2022 at 21:41 | comment | added | Robert Bryant | OK. As I suspected, the three-dimensional examples easily generalize to all higher dimensions as a 3-parameter family of non-trivial gradient almost Ricci solitons with harmonic curvature. They are not completely explicit, though, because each equivalence class of solutions corresponds to an integral curve of a vector field in $\mathbb{R}^3$. That vector field depends on a parameter: the (constant) scalar curvature $S=n(n{-}1)c$ of the metric $g$. I don't know how to integrate the vector field in elementary terms, but in the case $c=0$ phase portrait methods give good qualitative information. | |
| Nov 27, 2022 at 18:45 | history | edited | Matheus Andrade | CC BY-SA 4.0 |
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| Nov 27, 2022 at 18:43 | comment | added | Matheus Andrade | @RobertBryant thanks a lot! However, I must apologize again for not having included the hypothesis of dimension $\geq 4$ in my original post when I mentioned "part of some work that I've been doing with some other people...". But I am indeed very interested in knowing the explicit example of this $3$-parameter family you mentioned, I would very much appreciate it if you shared it. | |
| Nov 27, 2022 at 14:06 | comment | added | Robert Bryant | I thought a little bit about this and did a few caculations. In dimension 3, at least, it is not true that a gradient almost Ricci soliton with harmonic curvature is locally Einstein. There is a $3$-parameter family of mutually non-isometric, nontrivial examples $(g,f,\lambda)$ in dimension $3$ with the function $\lambda$ and the sectional curvatures not being constant. I haven't looked at higher dimensions, but I don't see why it would necessarily fail there. | |
| Nov 2, 2022 at 16:42 | history | edited | Matheus Andrade | CC BY-SA 4.0 |
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| Nov 2, 2022 at 16:41 | comment | added | Matheus Andrade | @RobertBryant sorry, I indeed forgot to add it as a hypothesis. I'll fix it, thanks for pointing it out. | |
| Nov 2, 2022 at 14:33 | comment | added | Robert Bryant | I'm curious why you define the condition of 'harmonic curvature' in one sentence and then don't refer to again. Is this a stray sentence that should be deleted or did you intend to add this as a hypothesis somewhere and forgot? | |
| Nov 2, 2022 at 0:06 | history | edited | Matheus Andrade | CC BY-SA 4.0 |
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| Nov 2, 2022 at 0:04 | answer | added | Jeffrey Case | timeline score: 4 | |
| Nov 1, 2022 at 22:13 | history | edited | Matheus Andrade | CC BY-SA 4.0 |
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| Nov 1, 2022 at 22:02 | history | edited | Matheus Andrade | CC BY-SA 4.0 |
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| Nov 1, 2022 at 21:55 | history | asked | Matheus Andrade | CC BY-SA 4.0 |