Can anyone help me solve the following exercise:
Let $ X $ be a killing field over a $ M $ Riemannian manifold. Prove that if $ p \in M $ is a critical point of the $ f = || X || ^ 2 $ function, then the integral curve of $ X $ passing $ p $ is a geodesic.
A $ X $ field is a killing field if $ L_ {X} <;> = 0 $. There are results that are equivalent to this definition.
Recall for a Killing field $X$, we have $\langle \nabla_AX, B\rangle+\langle \nabla_BX, A\rangle=0$. In particular, $X|X|^2=2\langle\nabla_XX, X\rangle=0$, i.e. $|X|^2$ is constant along any $X$-integral curve. So if $p$ is a critical point of $|X|^2$, then for every $t$, $\gamma(t)$ is also a critical point of $|X|^2$; here $\gamma$ is the integral curve of $X$ passing $p$. So at any $\gamma(t)$, we see $0=A|X|^2$ for any vector $A$, that is, $2\langle \nabla_AX, X \rangle=0$. So $\langle \nabla_XX, A\rangle=-\langle \nabla_AX, X\rangle=0$. Since $A$ is arbitrary, we see $\nabla_XX=0$, i.e. $\gamma$ is a geodesic.