Suppose $(M,g)$ is a Riemannian manifold, $\nabla$ the Levi-Civita connection and $\{e_i\}$ a basis of unit vector in in the tangent space $T_p M$ at $p\in M$.
Take now the geodesic $\gamma(t)$ verifing $\gamma(0)=p$ and $\dot{\gamma}(0)=e_0$, and take a variation $\Gamma(s,t)$ of $\gamma(t)$ through geodesics. We define a frame on the variation in the following way: parallel transport $\{e_i\}$ along the transverse curve $\Gamma(s,0)$, and then, for each $s$, parallel transport along $\Gamma(s,t)$. Let $\{e_i^{(s,t)}\}$ denote this frame on the point $\Gamma(s,t)$.
Now, I want to compute the covariant derivative of, for example, $\nabla_{e_1^{(0,t)}} e_1^{(0,t)}$. Is it possible to relate this with $\nabla_{e_1^{(0,0)}} e_1^{(0,0)}$?
What I tried: since the covarint derivative determines the covariant differentitation, and thanks to the way we have constructed the frame, we can write:
$$ \nabla_{e_1^{(0,t)}} e_1^{(0,t)}=\lim\limits_{s\rightarrow 0}\frac{P_{(s,t)}^{(0,t}P_{(s,0)}^{(s,t)}P_{(0,0)}^{(s,0)}P_{(0,t)}^{(0,0)}e_1^{(0,t)}-e_1^{(0,t)}}{s}. $$ This expression make me think of the relation between the Riemannian curvature tensor and the Levi-Civita connection, but since the limit is not taken on $t$, I am confused. Any ideas?