Timeline for Locally Riemannian Connection
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 18, 2018 at 11:51 | vote | accept | Aureliano Skirzewski | ||
| Jan 17, 2018 at 14:36 | answer | added | Robert Bryant | timeline score: 15 | |
| Jan 15, 2018 at 14:46 | comment | added | Aureliano Skirzewski | The condition $g_{ap}R^p{}_{bcd} + g_{bp}R^p{}_{acd} = 0$ can be rewritten as $\delta^{(p}_{(a}R^{q)}{}_{b)cd} g_{pq} = 0,$ a collection of matrices labeled by $c$ and $d$ with the same kernel... is there a way to identify the existence of a common eigenvector with null eigenvalue? | |
| Jan 13, 2018 at 21:32 | comment | added | Deane Yang | It suffices to but doesn't appear to be easy to solve the following two equations: $\partial_cg_{ab} = g_{ap}\Gamma^p_{cb} + g_{bp}\Gamma^p_{ac}$ and $g_{ap}R^p{}_{bcd} + g_{bp}R^p{}_{acd} = 0$. Indeed, additional conditions on the curvature tensor and its covariant derivatives are probably needed. | |
| Jan 13, 2018 at 21:29 | comment | added | Deane Yang | Sorry, but I didn't think this through. Now that I've done a little bit of calculation, this looks like an overdetermined system of PDEs that isn't easy for me to analyze. | |
| Jan 12, 2018 at 18:41 | comment | added | Aureliano Skirzewski | I really want to understand that argument @DeaneYang, could you suggest some literature? or explain a little more... I do not understand that article about parallel sections | |
| Jan 12, 2018 at 17:39 | comment | added | Ben McKay | You could employ the holonomy group in an arbitrarily small open set, which would already give a nontrivial condition, locally but not infinitesimally. | |
| Jan 12, 2018 at 17:34 | comment | added | Deane Yang | Note that $dg_{ab} = (g_{ap}\Gamma^p_{bc} + g_{bp}\Gamma^p_{ac})\,dx^c$, so you can use the Frobenius theorem. | |
| Jan 12, 2018 at 17:31 | comment | added | Aureliano Skirzewski | Within that article, Richard Atkins (the author) argues that the local problem has been previously solved by means of constructing a derived flag and refers to arxiv.org/pdf/0804.1732.pdf . This last article says "We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the Frobenius Theorem." Is that the question I made? | |
| Jan 12, 2018 at 17:16 | comment | added | j.c. | That question is not the same as yours, but unless I am misunderstanding something, I believe several of the answers and references will be helpful to you. See e.g. arxiv.org/abs/0804.2698 cited in John's answer mathoverflow.net/a/173399 | |
| Jan 12, 2018 at 17:03 | comment | added | Aureliano Skirzewski | That question is concerned with the existence of a metric, is that strictly the same question? | |
| Jan 12, 2018 at 16:59 | comment | added | j.c. | This question is related mathoverflow.net/questions/54434 | |
| Jan 12, 2018 at 16:51 | comment | added | Aureliano Skirzewski | Nevermind the existence of a unique metric tensor on the whole manifold | |
| Jan 12, 2018 at 16:43 | comment | added | Aureliano Skirzewski | Non local information as the holonomy group is not accessible so I wish to restrict to what can we know locally | |
| Jan 12, 2018 at 16:42 | history | asked | Aureliano Skirzewski | CC BY-SA 3.0 |