Local present of $\mathcal{L}$-geodesic

Question

$X=\dot{\gamma}(\tau)$. I want to locally present the $\mathcal{L}$-geodesic , let $X=X^i e_i$, $\{e_i\}$ is the basis in a local coordinate. So, $$ \nabla_XX=X^iX^j_ie_j +X^iX^j\Gamma_{ij}^ke_k \\ \nabla R =g^{ij}\partial_iR e_i \\ \frac{1}{2\tau}X=\frac{1}{2\tau}X^i e_i \\ Ric(X,\cdot)=X^iR_{ij}dx^j $$

As I know the 3.2.1 is a tensor equation, but the $Ric(X,\cdot)$ is not a vector, how to understand it ?

Answer 1

In a local coordinate, the 3.2.1 can be presented as $$ u_{\theta\theta}^k+u_\theta^iu_\theta^j\Gamma_{ij}^k-\frac{1}{2}g^{jk}R_{,j} +\frac{u_\theta^k}{2\tau}+u_\theta^iR_{ij}g^{jk}=0 $$ In fact ,$2Ric(X,\cdot)$ should be $2g^{-1}Ric(X,\cdot)$. I use $\theta$ as the parameter of curve i.e. $\theta=\tau$.