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?