Talk:Covariant derivative

curvature, parallel transport, geodesics

I removed the last two subsections (curvature and parallel transport-geodesic) they do not add anything to the correspondent articles and badly written.

The subsection on Levi-Civita connection is moved to Fundamental theorem of Riemannian geometry.

(I just realized that part of it can be used in lie bracket, but will do it next time) Tosha 20:47, 20 Jul 2004 (UTC)

∇ vs D

I think to change from D to ∇, it will make it consistent with other articles. Tosha 13:24, 21 Jul 2004 (UTC)

I agree that we should stick to one standard. I'm going to edit the page for spelling/grammar in a moment... - Gauge 02:50, 3 Aug 2004 (UTC)

A little more editing needed here

The section Informal definition using an embedding into Euclidean space ends its last mathematical statement as follows:

"... and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric:" $$g_{kl}{\Gamma ^{k}}_{ij}={\frac {1}{2}}\left({\frac {\partial g_{jl}}{\partial x^{i}}}+{\frac {\partial g_{li}}{\partial x^{j}}}-{\frac {\partial g_{ij}}{\partial x^{l}}}\right).$$

But whoever wrote that only provided an equation that the Christoffel symbols satisfy.

That it not the same as "yielding" the Christoffel symbols.

It seems to me that it will be helpful to use the inverse matrix (g_kl)^-1 of (g_kl) to try to isolate the Christoffel symbols.

I hope someone knowledgeable about this subject can fix this, so we really have a definition of each individual Christoffel symbol, isolated on the left side of an equation, in terms of the derivatives of the metric tensor and other things.

Fixed equation 1.

In the section "Vector fields", I have fixed equation 1. so that it is not written with vectors (or vector fields) *immediately to the left* of functions that the vector fields do not operate on.

Instead, the functions just multiply the vectors. In order to write that unambiguously, the functions need to be on the left of the vectors.

if $g$ is nondegenerate...

Added form for Christoffel symbol of the 2nd kind, for nondegenerate $g$. I would add that the order of the indices doesn't match what I've read elsewhere. This starts to matter, I suggest, when an asymmetric metric is considered, for whatever reason. Ric.Peregrino (talk) 20:11, 2 May 2024 (UTC)