Let $X$ be a Killing field on a Riemannian manifold $M$, and let $\nabla$ denote the Levi-Civita connection. Define $f: M \rightarrow \mathbb{R}$ by $f(q) = \langle X, X \rangle_q$, and let $p$ be a critical point of $f$, so that $df_p = 0$.
Do Carmo's hint for 5.2(b) implies that for all $Z$ in $\mathcal{X}(M)$ (the set of smooth vector fields on $M$), $\langle \nabla_X \nabla_Z X, Z \rangle = -\langle \nabla_Z X, \nabla_X Z \rangle$. He seems to derive this via an application of the "Killing equation" $\langle \nabla_W X, V \rangle + \langle \nabla_V X, W \rangle = 0$, which holds for all $W, V \in \mathcal{X}(M)$ whenever $X$ is Killing.
However, the Killing equation would only seem to apply to $\langle \nabla_X \nabla_Z X, Z \rangle$ if $\nabla_Z X$ was Killing. Is the equation $\langle \nabla_X \nabla_Z X, Z \rangle = -\langle \nabla_Z X, \nabla_X Z \rangle$ correct, and if so, why? For the claim of the exercise it suffices to show the equation holds at $p$.
Best Answer
Yes since
$$\langle \nabla_X \nabla_Z X, Z \rangle = X \langle \nabla_Z X, Z \rangle -\langle \nabla_Z X, \nabla_X Z \rangle$$
Now, apply the Killing equation when $W=V=Z$ to get $\langle \nabla_Z X, Z \rangle=0.$