Intrinsic derivative of constant vector field along a curve

1. Jan 24, 2017

harmyder

1. The problem statement, all variables and given/known data
Suppose that $T_i$ is the contravariant component of a vector field $\mathbf{T}$ that is constant along the trajectory $\gamma.$ Show that intrinsic derivative is $0.$

2. Relevant equations

$$\frac{\delta T_i}{\delta t} = \frac{dT^i}{dt}+V^j\Gamma^i_{jk}T^k$$

3. The attempt at a solution

\begin{align}\mathbf{T} = T^i \mathbb{Z}_i\\T^i = \frac{d\mathbf{T}}{dZ_i}\label{ti}\end{align}
But from $\ref{ti}$ i see that $T_i=0.$ Probably, $\ref{ti}$ is wrong.

Another attempt:)
\begin{align} \mathbf{T} &= T^i \mathbb{Z}_i\\ \mathbf{T}\cdot\mathbb{Z}^i &= T^i\\ \frac{dT^i}{dt}&= \frac{d\mathbf{T}}{dt}\mathbb{Z}^i + \mathbf{T}\frac{\partial\mathbb{Z}^i}{\partial Z^j}\frac{dZ^j}{dt}\\ &=-\mathbf{T}\Gamma^i_{jk}\mathbb{Z}^k\frac{dZ^j}{dt}\\ &=-\mathbf{T}\mathbb{Z}^k\Gamma^i_{jk} V^j\\ &=-T^k\Gamma^i_{jk}V^j \end{align}

OMH, looks like i have solved it while writing it here. Just need a confirmation.

Last edited: Jan 24, 2017