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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)

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What definition of orientability do you want to use? FYI The manifold $M(e)$ is a special-case construction of what people like to call "the orientation cover" of a manifold. –  Ryan Budney Nov 9 '10 at 23:26

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up vote 3 down vote accepted

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)

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