TITLE: Killing vector fields restricted to geodesics
QUESTION [6 upvotes]: Given a Riemannian manifolds $(M,g)$, a Killing vector field $X$ on $M$, and a geodesic $\gamma: K \rightarrow M$ defined on an interval $K \subseteq \mathbb{R}$, how does one show that $X \circ \gamma$ is a Jacobi field along $\gamma$?

REPLY [2 votes]: A vector field $X$ is Killing if and only if it generates a flow $\phi_s$ by isometries. We obtain a geodesic variation $\Gamma(s,t)=\phi_s \gamma(t)$, and 
$$ \frac{\partial \Gamma}{\partial s}\bigg|_{s=0} = \frac{ \partial \phi_s \gamma(t) }{\partial s} \bigg|_{s=0} =X_{\gamma(t)} $$ 
is a Jacobi field.