Let $D\subseteq TM$ a totally geodesic distribution of $M\subseteq \mathbb{R}^{n+p}$ (with the induced metric), and $\xi$ unit and normal to $M$, such that $R^\perp(X,Y)\xi=0$ for $X,Y\in D$, is this enough information to guarantee that $\xi$ is $D$-parallel? In other words, this implies that $\nabla^\perp_X \xi=0$ for all $X\in D$?
My idea is to fix a point and use the parallel transport on curves such that their velocity are in $D$, to define a normal vector field wich dosent depends on the path by the curvature condition (for simplicity lets assume simply connected), but im not able to prove that this new vector coincides with $\xi$
Any suggestion? it this even true? Thanks for any future help!