I want to understand when a given Killing vector field $X$ in a manifold $M$ is Parallel.
I know that when $X_p$ is zero and $\nabla_v X = 0$ for every $v \in T_pM$ we have that $X=0$. What can we say when $X_p$ is not zero and we still have $\nabla_v X = 0$ for every $v \in T_pM$? Maybe with more hypothesis we can guarantee that X is parallel.
I also know that when we restrict X to a geodesic we get a Jacobi vector field along the geodesic, maybe I can use that fact to calculate covariante derivatives.