$M$ is a Riemannian manifold with Riemannian connection $\nabla$. $X$ is a vector field on $M$. It is not in general true that $\nabla _XX=0$, for example a geodesic $\gamma$ satisfies $\nabla_{\frac{d\gamma}{dt}}\frac{d\gamma}{dt}\neq0$.
(i) What are some nice conditions on $X$ that garantee $\nabla _XX=0$ ( for example, $X$ is Killing)?
(ii) Take orthonormal fram ${E_i}$. Is it true that $\nabla_{E_i}E_i=0$?