Timeline for Covariant Derivative of sections of a pullback bundle
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 30, 2020 at 18:21 | comment | added | Quarto Bendir | I don't know of anywhere where I find it comprehensible. It is essentially inside of volume 1 of Kobayashi and Nomizu's "Foundations of Differential Geometry". You could also check Eells and Sampson's article "Harmonic mappings of Riemannian manifolds" and Eells and Lemaire "A report on harmonic maps" and "Selected topics in harmonic maps", maybe Schoen and Yau "Lectures on Harmonic Maps" | |
| Dec 30, 2020 at 15:17 | comment | added | shuhalo | @QuartoBendir: thank you. Still wondering whether this has been written down anywhere because it should. | |
| Dec 30, 2020 at 15:01 | comment | added | Quarto Bendir | It is also worth noting that the induced connection on $T^\ast M\otimes\phi^\ast TN$, which is somewhat more common, is given by $\nabla_i\omega_j^\alpha=\frac{\partial\omega_j^\alpha}{\partial x^i}-\Gamma_{ij}^p\omega_p^\alpha+\frac{\partial f^\beta}{\partial x^i}\Gamma_{\beta\gamma}^\alpha\omega_j^\gamma.$ | |
| Dec 30, 2020 at 14:59 | comment | added | Quarto Bendir | Yes, the former for $M$ and the latter for $N$. They could be any connections on $TM\to M$ and $TN\to N$, it does not matter if they are metric-compatible or torsion-free. | |
| Dec 30, 2020 at 14:08 | comment | added | shuhalo | @QuartoBendir: here you are using two sets of Christoffel symbols, right? | |
| Dec 30, 2020 at 10:47 | comment | added | Quarto Bendir | For #3, the formula in local coordinates is $\nabla_i\omega^{j\alpha}=\frac{\partial\omega^{j\alpha}}{\partial x^i}+\Gamma_{ip}^j\omega^{p\alpha}+\frac{\partial f^\beta}{\partial x^i}\Gamma_{\beta\gamma}^\alpha\omega^{j\gamma}.$ | |
| Dec 29, 2020 at 18:49 | comment | added | Deane Yang | I don't know any reference. As you say, it's usually easy to figure most of them out yourself. | |
| Dec 29, 2020 at 18:46 | comment | added | shuhalo | @DeaneYang: Do you know any exposition of, broadly speaking, natural constructions with different connections? While I believe I can guess most formulas from this point on, it would be great to have a citable reference. | |
| Dec 27, 2020 at 17:56 | comment | added | Deane Yang | I agree this is not standard textbook material, even though it should be. My answer on the page I cited is a terse summary of the situation for the pullback bundle and connection. This shows that the connection on $M$ plays no role in the definition of the connection on $\phi^*TN$. As for the tensor product, this is a separate independent step. If you have two vector bundles with connections, then the tensor product has a natural connection given by the product rule: $\nabla_X (\sigma\otimes\tau)=(\nabla_X \sigma)\otimes\tau+\sigma\otimes(\nabla_X\tau)$. | |
| Dec 27, 2020 at 17:00 | comment | added | shuhalo | @DeaneYang: thanks. I reformulated my posting to clarify what I am looking for. I don't think this is standard textbook material. | |
| Dec 27, 2020 at 16:56 | history | edited | shuhalo | CC BY-SA 4.0 |
added third part of the posting
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| Dec 27, 2020 at 16:35 | comment | added | Deane Yang | Here’s a discussion about this. mathoverflow.net/q/49272/613 | |
| Dec 27, 2020 at 16:15 | history | asked | shuhalo | CC BY-SA 4.0 |