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I read that an oriented manifold is a manifold with a choice of orientations for each tangent space so that for $p \in M$, there is an open set $U$ and a collection of vector fields $X_1,...,X_n$ so that for all $q \in U$ $X_1(q),...,X_n(q)$, is a basis of $T_qM$ belonging to the orientation of $T_qM$. What does it mean for manifold to be "orientable"? Is it the same thing as oriented?

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Two related answers (full disclosure: by me)…,… – Neal Dec 6 '12 at 2:21

A manifold is orientable if there exists an orientation. A manifold is oriented if some orientation is fixed. In particular, every oriented manifold is orientable.

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An interesting characterization of orientability is as follows. For each manifold $M$ we can form the orientation double cover of $M$ usually denoted $M^{*}$, which is orientable by construction. $M$ is orientable iff $M^{*}$ is not connected. In the case where $M$ is not orientable, $M^{*}$ is homeomorphic to two disjoint copies of $M$.

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A manifold is orientable if there exists an atlas for it, such that the transition map between any two surface patches in the atlas has jacobian with positive determinant.

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