Timeline for Writing a Taylor series with covariant derivatives (connections)?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 23 at 4:31 | comment | added | R. Rankin | @BertramArnold Are you saying that jet bundles, (which I know a choice of cross section of is essentially a taylor series expansion) already incorporate a covariant derivative into the Taylor expansion? | |
| Aug 11, 2021 at 8:47 | comment | added | Bertram Arnold | It's the usual Taylor series in a local chart, but you re-express the partial derivatives via connection derivatives. The point is that the complicated transformation of iterated derivatives under a change of chart, known as Faà di Bruno's formula, can be absorbed into the transformation of the curvature form. For an intrinsic formulation which does not require a choice of connection, you might want to check out the formalism of jet bundles. | |
| Aug 11, 2021 at 5:07 | comment | added | Ma Joad | @BertramArnold So what's the form of the series? | |
| Aug 10, 2021 at 7:37 | comment | added | Bertram Arnold | You can pick a torsion-free connection $\nabla^{TX}$ on $TX$ and consider the iterated connection derivatives $\nabla^{T^*X\otimes\dots\otimes T^*X\otimes E}\circ\dots\circ\nabla^{T^*X\otimes E}\circ \nabla^E:\Gamma(E)\to \Gamma((T^* X)^{\otimes n}\otimes E)$. Note that the antisymmetrization of any two of the $n$ factors of $T^*X$ in the result can be expressed via covariant derivatives of the curvature tensors of $\nabla^{TX}$ and $\nabla^E$. The projections $\Gamma(E)\to \Gamma((\operatorname{Sym}^n T^*X)\otimes E)$ are then the correct generalization of the Taylor coefficients. | |
| Aug 10, 2021 at 2:31 | history | edited | Ma Joad | CC BY-SA 4.0 |
added 237 characters in body
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| Aug 10, 2021 at 1:47 | history | asked | Ma Joad | CC BY-SA 4.0 |