Timeline for What is the Levi-Civita connection trying to describe?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 6, 2020 at 16:22 | comment | added | Ben McKay | You could allow second derivatives along with the first derivatives and still only uncover the Levi-Civita connection. The difference between connections is a 3-tensor, so cannot be reflection invariant in geodesic normal coordinates, while the quadratic terms in the Taylor series of a metric must be invariant under reflection, because any parabola is reflection invariant around its focal line. | |
| Nov 17, 2020 at 0:38 | comment | added | Tom Mrowka | This was a beautiful answer. When is Bryant's intro to Riemannian (and other) geometry coming out? | |
| Nov 16, 2020 at 2:09 | history | edited | Robert Bryant | CC BY-SA 4.0 |
added 5 characters in body
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| Nov 16, 2020 at 2:02 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Added a remark about connections compatible with g and having the same geodesics as its Levi-Civita connection.
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| Nov 15, 2020 at 20:53 | comment | added | Robert Bryant | @DavidESpeyer: I realize, on re-reading it, that my first response to your comment was not the best. I should have acknowledged that your construction in dimension 3 (with an orientation added to the data) is an interesting one, and one that I hadn't considered. Thanks for pointing it out, and thanks for your kind words! | |
| Nov 15, 2020 at 18:43 | comment | added | Robert Bryant | @DavidESpeyer: Using an orientation is using more than the metric. I think my answer stands correct as it is. Also, even if you add orientation, you won't get any new examples other than in dimension $3$. | |
| Nov 15, 2020 at 18:14 | comment | added | David E Speyer | Regarding the first paragraph: It seems to me that, on an oriented Riemmannian $3$-fold, there is another natural construction that only uses the metric and its first derivative (and the orientation). Using the orientation and the metric, we can define a cross product on each tangent space. For any scalar $c$, we can define $\nabla_X(Y) = \nabla^{LC}_X(Y) + c X \times Y$, where $\nabla^{LC}$ is the Levi-Cevita connection. (This nitpick aside, great answer!) | |
| Nov 15, 2020 at 16:34 | comment | added | Deane Yang | Robert, I think what you say in your first paragraph is how the Levi-Civita connection should be introduced, rather than having the symmetry condition be pulled out of thin air. To me it better motivates the definition. | |
| Nov 15, 2020 at 15:10 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Fixed some typos and cleaned up some language
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| Nov 15, 2020 at 13:52 | history | answered | Robert Bryant | CC BY-SA 4.0 |