Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer

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 $M$.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.