Timeline for Are differential geometric and physics conventions for covariant derivatives consistent?
Current License: CC BY-SA 4.0
9 events
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| Sep 26, 2021 at 14:04 | answer | added | Vincent Thacker | timeline score: 1 | |
| Sep 26, 2021 at 10:51 | comment | added | glS | @VincentThacker that is my understanding as well. In fact, in the linked lectures, they say that $\nabla_a\vec S_b$ is normal to the surface, which seems in direct contrast with the covariant derivative as an object which by definition gives tangent vectors as output. This makes me think that there is a different way to understand "covariant derivative" in this context, which is pretty much what I'm asking. Still, I've seen these rules of how $\nabla_\alpha$ acts on objects with raised/lowered indices quite often in physics literature, and formulas like (4) seem a direct consequence of those | |
| Sep 26, 2021 at 10:46 | comment | added | Vincent Thacker | Alright, your equation (4) is evidently false. There is no reason, in general, for the covariant derivative of a vector to have zero dot product with all basis vectors. That will only be true if it were the zero vector (or normal to the surface). Either the notation is incorrect or you confused quantities defined on the manifold with those defined on the ambient (Euclidean) space. | |
| Sep 26, 2021 at 10:32 | history | edited | glS | CC BY-SA 4.0 |
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| Sep 26, 2021 at 10:27 | history | edited | glS | CC BY-SA 4.0 |
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| Sep 26, 2021 at 10:24 | comment | added | glS | @VincentThacker I'm not completely sure what's the formalisation of the ideas underlying (3) and (4) here; that is, de facto, part of what I'm asking here I think. My understanding is that "embedded" is understood as you say, yes; "standard derivative" should mean the directional derivative taken in the embedding space (which is I think also the same as what you say). So what I write with $\partial_\alpha\vec S_\beta$ here | |
| Sep 26, 2021 at 9:59 | history | edited | Qmechanic♦ |
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| Sep 26, 2021 at 9:49 | comment | added | Vincent Thacker | By "embedded surfaces", do you mean a manifold embedded in Euclidean space? Also, by "standard derivative", do you mean the canonical connection in Euclidean space arising from the natural isomorphism? | |
| Sep 26, 2021 at 9:41 | history | asked | glS | CC BY-SA 4.0 |