Prove the unique existence of a unit vector field in an orientable Riemannian manifold with boundary. Let $M$ be an orientable Riemannian manifold with boundary, prove the unique existence of a unit vector field $X$ on $M$, such that for any point $p\in\partial M$, $X_{p}$ is inward-pointing and orthogonal to $T_{p}(\partial M)$.
I'm not sure whether my idea is right, but I guess by solving some ordinary partial equations we may get the unique existence, but I don't how to build them, and I don't know how to use the condition of Riemann metric, so can someone give me some hints? thanks.
 A: Since there is no condition imposed at interior points of $M$ there is no reason for such a vector field to be unique.  For example, on a half-cylinder (with boundary $S^1$) you could construct tons of such vector fields.  If you meant that the value of the vector field along the boundary is unique, this follows from the fact that there is a unique normal direction at each point of $T_p(\partial M)$ (since the "opposite" direction points outward rather than inward).  This question seems more elementary than looking at ODEs.
For example on the half-cylinder $M$ one can choose any smooth function $\theta(p), p\in M$ which vanishes along the boundary, and modify any given vector field $V(p)$ by rotating $V(p)$ by angle $\theta(p)$ so one can't possibly expect uniqueness.
Also, in general there may not be such a vector field on all of $M$.  For example, on the unit disk there is no nonvanishing vector field extending the inward normal along the boundary.  This easily follows from Brouwer's theorem.
