In Petersen's Riemannian geometry page 118 he computes the curvature operator on the product Riemannian manifold $S^n(1/a) \times S^m(1/b)$. For my question the fact that it is a product of spheres does not seem relevant - it could be any product manifold. In his computation he lets $Y$ be a unit vector field on $S^n$, $V$ a unit vector field on $S^n$ and $X$ be a unit vector field on either $S^n$ or $S^m$ that is perpendicular to $Y,V$. He then computes that $g(\nabla_YX,V) = 0$ via the Koszul formula. This all seems fine to me.
The part where I'm having trouble is he then claims that $g(\nabla_YX,V) = 0$ implies that $\nabla_YX = 0$ if $X$ is tangent to $S^m$ and $\nabla_YX$ is tangent to $S^n$ if $X$ is tangent to $S^n$. I'm not exactly sure how this result follows from $g(\nabla_YX,V) = 0$? It seems like maybe he is using the fact that the Levi-Cevita connection for a product manifold is essentially the sum of the respective connections when acting on vectors fields tangent to the manifolds in the product, but if so, why the discussion on $g(\nabla_YX,V) = 0$?