Show that there exists a vector field $V$ defined on $M$ such that $V_p$ spans $D_p$ for all $p \in M$.
I don't usually ask for "hint to start" but I am quite stumped by this problem.
The only thing I can find that's relevant, is possibly using the orientation covering or something similar for $M$, since we know that all simply connected smooth manifolds are orientable.
The orientation cover is a fiber bundle whose preimage of every point in $p$ are two orientation of $T_p M$, and I can see how we may be able to imitate the construction for the orientation cover to construct a fiber bundle from $D$ as there are only two points on $D_p$ that span $D_p$ but also have length $1$ given a Riemannian metric.
Edit: Changed $V_p \in D_p$ to $V_p$ spans $D_p$ for all $p \in M$.