If there exists a nontrivial vector field $V\not=0$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be flat?

That is, if a Riemannian maniflod exists a vector field $V$ parallel transport along any vector field, then is this maniflod flat?

everhold? $\endgroup$ – Mariano Suárez-Álvarez Apr 20 '14 at 4:57