Say we have a Riemannian manifold $(M, g)$ with vector field $Y$, obeying:
- $g(Y, Y) = 1$; and
- the $1$-form $\varphi(X) = g(X, Y)$ is $d$-closed, $d\varphi = 0$.
I know that the integral curves of $Y$ are geodesics, i.e. $D_Y Y = 0$. Does it follow that these geodesics are locally orthogonal to a family of hypersurfaces $f = k$?