What is orientation?

An orientation of $M$ is a choice, for each $p \in M$, of one of the two orientation classes of bases of the tangent space $T_pM$, which is locally coherent: around each point there is a parametrization $\gamma$ whose coordinate frame $(\partial_1\gamma, \dots, \partial_k\gamma)$ is positively oriented at every point of its domain. Such parametrizations are called direct. $M$ is orientable if an orientation exists; the Möbius band shows this can fail. All submanifolds in this chapter are oriented.