Parallel Transport of a Tensor: Understand Equation

[AndersF]

TL;DR Summary

Understanding the equation for a tensor to be parallel-transported

According to my book, the equation that should meet a vector $\mathbf{v}=v^i\mathbf{e}_i$ in order to be parallel-transported in a manifold is:

$v_{, j}^{i}+v^{k} \Gamma_{k j}^{i}=0$

Where $v_{, j}^i$ stands for $\partial{v^i}{\partial y^j}$, that is, the partial derivative of the component $v^i$ of $\mathbf{v}$ with respect to the general coordinate $y^j$. I see that there is a sum in $k$ form 1 to $n$, and that this equation must be meet for all $i,j=1,2,...,n$, being $n$ the dimenssion of the manifold.

However, I find it difficult to understand how to read this formula describing the condition for parallel transport of a tensor:

$T_{j_{1} j_{2} \ldots j_{r}, k}^{i_{1} i_{2} \ldots i_{s}}+\sum_{m=1}^{s} T_{j_{1} j_{2} \ldots j_{r}}^{i_{1} i_{2} \ldots p_{m} \ldots i_{s}} \Gamma_{p_{m} k}^{i_{m}}-\sum_{n=1}^{r} T_{j_{1} j_{2} \ldots q_{n} \ldots j_{r}}^{i_{1} i_{2} \ldots i_{s}} \Gamma_{j_{n} k}^{q_{n}}=0$

(My theory is that whoever wrote that formula probably did so to engage in a competition of convoluted mathematical notations... )

Could somebody please help me understand it how should be read? For example, how would it apply for a tensor of order three $T^{a,b}_{\alpha,\beta}$?

 

[ergospherical]

It's not the easiest notation to read is it, haha. A tensor $T^{\mu \dots}_{\nu \dots}$ is parallel transported along a curve of tangent $u^{\mu} = dx^{\mu}/d\lambda$ if \begin{align*}
\dfrac{DT^{\mu \dots}_{\nu \dots}}{d\lambda} = u^{\rho} \nabla_{\rho} T^{\mu \dots}_{\nu \dots} = u^{\rho} (\partial_{\rho} T^{\mu \dots}_{\nu \dots} + \Gamma^{\mu}_{\sigma \rho} T^{\sigma}_{\nu} - \Gamma^{\sigma}_{\nu \rho} T^{\mu \dots}_{\sigma \dots} + \dots) = 0
\end{align*}There's one correction term per tensor index in the covariant derivative. Notice the patern: each index is pulled onto the Christoffel symbol and then replaced with a dummy index. Terms correcting for upper indices appear with a $+$ sign, and terms correcting for lower indices appear with a $-$ sign.

 

[AndersF]

Oh ok, it is by far much clearer the way you wrote it. Now I see it, thanks!