Trouble understanding ##g^{jk}\Gamma^{i}{}_{jk}##

[shooride]

Hi friends,

I'm learning Riemannian geometry. I'm in trouble with understanding the meaning of $g^{jk}\Gamma^{i}{}_{jk}$. I know it is a contracting relation on the Christoffel symbols and one can show that $g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})$ using the Levi-Civita connection, etc-- rhs is similar to the divergence of an antisymmetric tensor field (!)--. But, how should one interpret and call $g^{jk}\Gamma^i{}_{jk}$?! I'm interested in this term since it appears in the Laplacian of the function $f$ (Laplace–Beltrami operator). Especially, $\Delta f=div ~\nabla f= \partial^2 f-g\Gamma\partial f$ (not using the Einstein notation). The interpretation of first term in the rhs is clear; Is there any clear/simple interpretation for second term?!

 

[shooride]

I guess I should name $g^{jk}\Gamma^i{}_{jk}$, $a^i$.

 

[strangerep]

First, do you understand why the covariant derivative of a vector has 2 terms? (I.e., a partial derivative and a term involving $\Gamma$) ?

 

[shooride]

First, do you understand why the covariant derivative of a vector has 2 terms? (I.e., a partial derivative and a term involving $\Gamma$) ?

Yeah, since the covariant derivative is a covariant . I think I was a bit confused when I asked this question . Anyway, do you know whether there is a particular name for the identity $g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})$?!

 

[strangerep]

[...] do you know whether there is a particular name for the identity $g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})$?!

No, I don't know a name.

BTW, for Riemannian geometry stuff involving indices, (which mathematicians usually hate), you can sometimes get more answers by asking in the relativity forum.