Orientation

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Oriented manifolds

Two charts $(U, \phi)$ and $(V, \psi)$ on $M$ are orientation-compatible if they are compatible and the transition map $\psi \circ \inv \phi$ is orientation-preserving on $\R^m$, i.e.

\[\det(D(\psi \circ \inv \phi)) > 0 \text{ on } \phi(U \cap V)\]

An oriented atlas is an atlas where any two coordinate charts are orienation-compatible.

A manifold $M$ is orientable if it admits an oriented atlas.

Maximal oriented atlases

Given an oriented atlas $\A$, we can find a maximal oriented atlas $\overline A$ which is the set of all coordinate charts that are orientation-compatible with $\A$.