Timeline for What is the Levi-Civita connection trying to describe?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Nov 16, 2020 at 21:41 | comment | added | Chris Gerig | @RobertBryant Oops, Volume II. I’ll email you the 4-volumes in pdf. | |
| Nov 16, 2020 at 21:38 | comment | added | Robert Bryant | @ChrisGerig: Thanks for this reference Chris. I lost my volumes of Spivak's *A Comprehensive Introduction to Differential Geometry" years ago (they were the old paperbound pre-TeX volumes) and, somehow never replaced them. I'll see if I can track down a local copy. | |
| Nov 16, 2020 at 19:56 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Nov 16, 2020 at 12:39 | answer | added | Jonathan Manton | timeline score: 18 | |
| Nov 16, 2020 at 9:12 | comment | added | Ben McKay | @Qfwfq: The notation $XY$ means the vector field $X^j (\partial Y^i/x^j)\partial/\partial x^i$. | |
| Nov 16, 2020 at 5:22 | comment | added | Chris Gerig | @RobertBryant Spivak also treats this in his Volume III book (Chapter 6 "Addendum 1"), but I enjoyed your answer because it delves further! | |
| Nov 16, 2020 at 2:05 | comment | added | Robert Bryant | @AndrewNC: I just added a remark at the end of my answer that explains this. What you may have read claiming that all the $g$-compattible connections have the same geodesics as the Levi-Civita connection turns out to be wrong. In fact, in dimension $2$, the Levi-Civita connection is the only $g$-compatible connection that has the same geodesics as the Levi-Civita connection. In higher dimensions, though, there are more $g$-compatible connections that have the same geodesics as the Levi-Civita connection of $g$. See my answer for details. | |
| Nov 16, 2020 at 0:44 | comment | added | Andrew NC | That's great, you just identified one of my misunderstandings! $XY$ is in fact not a derivation, but $XY-YX$ is. What an interesting oddity that I have completely missed. I remain confused, though, by my main point. It does seem like the parallel transport defined by rotation based on the difference in the $y$ coordinate satisfies that it preserves the metric, but does not satisfy that straight lines are geodesics... | |
| Nov 15, 2020 at 20:09 | comment | added | Qfwfq | "...the usual (and therefore Levi-Civita) connection then $\nabla_XY$ is just $XY$" - I'm confused by your notation: what do you mean by $XY$? (the covariant derivative along a given vector (field) of a tensor field is still a tensor field (of the same type), but $XY$ seems to usually denote a second-order differential operator...) | |
| Nov 15, 2020 at 18:59 | comment | added | Andrew NC | I have edited to fix a typo and clarify my example, and what $p_1$ and $p_2$ are. | |
| Nov 15, 2020 at 18:58 | history | edited | Andrew NC | CC BY-SA 4.0 |
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| Nov 15, 2020 at 17:29 | comment | added | Andrew NC | I would love more about this perspective, @BenMcKay . I think that would be very instructive for me. I'm also surprised to see that no one contradicted me that preserving the metric doesn't fix the geodesics, I've seen that in many places, but as you can see here I think I provide a counter-example... | |
| Nov 15, 2020 at 17:24 | answer | added | Gabe K | timeline score: 12 | |
| Nov 15, 2020 at 13:52 | answer | added | Robert Bryant | timeline score: 50 | |
| Nov 15, 2020 at 8:32 | answer | added | Ben McKay | timeline score: 28 | |
| Nov 15, 2020 at 8:28 | history | became hot network question | |||
| Nov 15, 2020 at 8:25 | comment | added | Ben McKay | The condition that $\nabla_X Y=XY$ is not invariant under diffeomorphism. Vanishing of torsion is diffeomorphism invariant. There are no other tensor invariants of a connection which are linear and zero order and diffeomorphism invariant. | |
| Nov 15, 2020 at 4:27 | answer | added | Deane Yang | timeline score: 18 | |
| Nov 15, 2020 at 4:18 | comment | added | Andrew NC | It works to what end? If we are proving properties of the parallel transport corresponding to the Levi-Civita connection, what is the merit of that as an object of interest? Additionally, are you claiming that it is inherently non-intuitive, and that any effort to understand it intuitively is misguided? | |
| Nov 15, 2020 at 3:41 | comment | added | Anton Petrunin | Because it works | |
| Nov 15, 2020 at 0:24 | history | asked | Andrew NC | CC BY-SA 4.0 |