Orientation on a Manifold Let M  be an (n-1)-manifold in R^n . Let M(e)  be the set of end-points of normal vectors (in both directions) of length e  and suppose e  is small enough so that M(e) is also an (n-1)-manifold. Show that M(e)  is orientable (even if M  is not)

Each point $p$ of $M(\epsilon)$ comes from a point $q$ in $M$. Consider the vector $X$ field on $M(\epsilon)$ which on $p$ takes the value $\vec{qp}$. Then $N$ is a non-zero normal field on $M(\epsilon)$ (maybe one needs to consider the projection of $N$ onto the normal line to $M(\epsilon)$ at each point, but that projection is surely non-zero and, moreover, to normalize this projection)
