This question is from exercise 6.12, page 143, in do Carmo's Riemannian Geometry. I am trying to prove it following the hint.
Let $X$ be a Killing field on a riemannian manifold $M$. Let $N=\{p\in M; X(p)=0\}$. Choose a connected component of $N$, $N_k$, and b) tells us $N_k$ is a submanifold of $M$. Let $E_p=(T_pN_k)^\bot$ and let $V\subset M$ be a normal neighborhood of $p$. Set $$N_k^\bot=\exp_p\ (E_p\cap \exp_p^{-1}(V))$$ Since $X_t$ is an isometric diffeomorphism, $(dX_t)_p: E_p\to E_p$, then the book says $X$ is tangent to $N_k^\bot$. I am confused with the last step, how can we get $X$ is tangent to $N_k^\bot$?