Orientations

Abstract

The purpose of this chapter is to introduce a subtle but important property of smooth manifolds called orientation. This word stems from the Latin oriens (“east”), and originally meant “turning toward the east” or more generally “positioning with respect to one’s surroundings.” Orientations of manifolds generalize the idea of choosing which direction along a curve is considered “positive,” which rotational direction on a surface is considered “clockwise,” or which bases in 3 dimensions are considered “right-handed.” Manifolds in which it is possible to choose a consistent orientation are said to be orientable. After defining orientations, we treat the special case of orientations on Riemannian manifolds and Riemannian hypersurfaces. At the end of the chapter, we explore the close relationship between orientability and covering maps. Orientations have numerous applications, most notably in the theory of integration on manifolds, which we will study in Chapter  16.