Framework:
The following definition and theorem come from [1]
$\textbf{Definition}.$ Let $\gamma : [0,T] \rightarrow M$ be a smooth curve, and $\epsilon > 0$. A variation of $\gamma$ is a smooth map $F :[0,T]\times(−\epsilon,\epsilon)\rightarrow M$ such that $F(t,0) = \gamma(t)$ for all $t\in[0,T]$ and $F(t,s) = \gamma_s(t)$.
The variation field of F is the vector field $V(t) = \frac{\partial }{\partial s}F(t, 0)$ along $\gamma$.
We denote by $l(\gamma_s)$ the length of the curve $\gamma_s$.
$\textbf{Theorem}$ [First variation of length]
Let $\gamma :[0,T]\longrightarrow M$ be any unit speed admissible curve and $F(t,s)$ a smooth variation of $\gamma$. Then $$\frac{d}{ds}l\left(\gamma_s\right)(0)=-\int_{0}^{T}\left<V(t),\nabla_{\dot{\gamma}}\dot{\gamma}\right>dt+\left<V(T),\dot{\gamma}(T)\right>-\left<V(0),\dot{\gamma}(0)\right>.$$
$\textbf{Remark.}$ $M$ is a Riemannian manifold.
I would like to know if the version of the theorem of the first variation of length exists on homogeneous spaces, i.e $M=G/H$ (and its proof). I already checked the book: An introduction to Lie groups and the geometry of homogeneous spaces (Andreas Arvanitogeorgos) and did not find what I was looking for.
Any suggestion?
[1] John M Lee, Riemannian manifolds: an introduction to curvature, vol. 176, Springer New York, 2006.