Talk:Levi-Civita connection

Page contents not supported in other languages.
From Wikipedia, the free encyclopedia

Stub history[edit]

Why did you kill this page? -- Anon

The problem is the article as it was written doesn't make sense. The text was: " The unique torsion free connection, preserving a riemannian metric". What does that mean? What does this relate to? Is it physics? biology? fiction? Remember you are writing for a non-specialist audience.
A quick Google tells me this is something to do with mathematics. The Anome gave a great example of how to write a good article on mathematical topics:
"(Mathematician) invented the concept of X in 18xx to represent (squeezy-pully-twisty things). A simple example, using modern notation is (example). (Explain notation). The idea has now been generalized to (stuff), which has uses in (other fields of math and science). The idea of X can be formalized as follows: let there be objects X such that (notation). Then (notation) (notation) (notation)..." (from this talk page)
See also Wikipedia:WikiProject Mathematics
If this article has nothing to do with mathematics, then that shows the problem even more clearly. In a case like this it is usually thought better to remove the article rather than have a confusing sub-stub. But much better is to improve the article, and if you can do that it would be great! I hope you stick around and give it a go. Regards -- sannse 19:30 26 Jun 2003 (UTC)

Formal definition[edit]

This section needs to be reviewed. With a firm background in vector calculus, I cannot interpret what the two equations are trying to express. Please explain where the RHS and LHS are coming from, and be more explicit with notation. I can only assume that X(g(Y,Z)) is supposed to mean the scalar multiplication of the vector field X with the inner product of the fields Y and Z, but this yields a vector quantity, and the RHS, which is the sum of inner products, would be a scalar. 98.235.167.113 (talk) 18:26, 8 March 2012 (UTC)[reply]
It is common for Xf to represent the Lie derivative of the function f by the vector field X (so that Xf is a function). This is not the same as the product of f and X, which would be written fX and would be a vector. Thus X(g(Y,Z)) means the Lie derivative of the function g(Y,Z), and the equations in this section make sense. — Preceding unsigned comment added by 83.104.131.53 (talk) 11:16, 3 May 2012 (UTC)[reply]
To say "It is common ..." is perhaps a bit strong, especially in this encyclopedic context. The notation strikes me as highly context-specific, and since the term "Lie derivative" does not occur once in the article, it is appropriate to add something like "... where Xf denotes the Lie derivative of ...". I would go so far as to say that a more explicit notation such as should be used, plus the mention that this represents the Lie derivative. — Quondum 13:09, 3 May 2012 (UTC)[reply]
there is no need to introduce lie drivatives here. to make the notation more intuitive replace Xf by which is if necessary easily explained as df(X).Peter Grabs (talk) 06:28, 2 August 2018 (UTC)[reply]

Derivative along curve[edit]

i suggest to replace this sections by something like this, because the given definition is not applicable for vectorfields along curves (as i know that notion):


The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by .

Given a smooth curve on we consider vector fields along , i.e. smooth sections of the pullback bundle , and define their derivative by

.

For this pullback connection specializes to

.

In particular, is a vector field along the curve itself. If vanishes, the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.

-- Peter Grabs (talk) 19:17, 29 April 2012 (UTC)[reply]

Connection preserves metric[edit]

The expression does not make sense to me. Isn't the property of metric preservation exactly ? — Preceding unsigned comment added by 2003:6B:A1E:4801:4000:2B7E:EC01:A50 (talk) 10:06, 25 May 2015 (UTC)[reply]

if you have a notion of derivation for functions () and vectors (, e.g. Levi Civita of TM), you can carry it over to arbitrary tensors like by requiring the leibniz rule, i.e. , this defines and lets you read off the equivalence of the two definitions of metricity.Peter Grabs (talk) 06:56, 2 August 2018 (UTC)[reply]

Animation of parallel transport[edit]

The animation in the section on parallel transport indicates that it shows parallel transport of a vector under two different metrics. However, the metrics are simply both the flat metric, but in Cartesian and polar coordinates. I'm not sure how the right-side animation is supposed to be interpreted. If the axes still represent Cartesian coordinates then the animation is clearly wrong as parallel transport should be identical to that in the left image. If the coordinate axes are for really, then the parallel transport curve (unit circle) passes through the coordinate singularity at and secondly the length of the vector is not preserved with respect to the metric. Jeldering (talk) 22:18, 20 January 2016 (UTC)[reply]

This turned out to be an erroneous anonymous edit that I reverted. Still, I think this section is not clear and needs rewriting/improvement. I may have a go at it if I find time. Jeldering (talk) 14:18, 21 January 2016 (UTC)[reply]

I spent a long time thinking about this section and it still confuses me. Part of the problem is that we are not used to "see" non-euclidean metrics on the plane. Also, if I'm not mistaken, both metrics should be the same in S^1, which is the situation shown here. I would appreciate it if someone took the time to explain in more detail (possibly in charts) how the different parallel transports arise. Emilparole (talk) 13:08, 20 February 2018 (UTC)[reply]

Some other people edited this page and screwed up the metrics again. I've tried my best, with the limited time and understanding I currently have, to make it at least a little bit clearer that and that the second transport is for a completely different metric. It still needs a rewrite, though. 73.114.132.157 (talk) 02:56, 18 July 2018 (UTC)[reply]

Parallel transport example is wrong[edit]

First the whole discussion about a singularity in the origin is besides the point, because both have this singularity, the big scary formula although not wrong is distracting and pointless and should be removed. The error is in the claim that both connections are "different" Levi-Civita connections, the one on the left is indeed a Levi-Civita connection but the one on the right is NOT! (this error is also present on the file page of the animation.) The animation on the right has a connection but it's not a Levi-Civita connection, i.e. it's non-symmetrical in the last two indices. The connection can be found by using the general formula for the Connection coefficients in a nonholonomic basis using the formula given in this article will result in all zero coefficients and parallel transport that does not change the vector and so identical to the animation on the left. --DelftUser (talk) 15:24, 25 February 2020 (UTC)[reply]

The metric used in the right image is wrong! the right image does not use the coordinates basis and so the metric can't be written as . Here is how the title & captions of the two animations should be:

Parallel transports under different connections
Cartesian transport
This transport is in the coordinates basis: with the Levi-Civita connection of the metric .
Polar transport
This transport is in the natural local tetrad: and the associated Ricci Rotation Coefficients.

Because the basis vectors are different it's wrong to write the metric with (), so this is my fix; I'll wait a while for comments before changing the article.--DelftUser (talk) 09:31, 29 February 2020 (UTC)[reply]

The example is correct, interpreting r and theta as functions on R^2. The former defines the standard metric on R^2 and the latter defines an incomplete metric on R^2-{(0,0)} which is isometric to the standard circular cone. I think it should be explained more clearly though, and there is probably a better example to use. Gumshoe2 (talk) 23:04, 5 January 2022 (UTC)[reply]