Orientations
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
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Orientations
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.” Thus, an orientation of a line or curve is a simply a choice of direction along it. As we saw in Chapter 11, the sign of a line integral depends on a choice of preferred direction along the curve. Mathematicians have extended the sense of the word “orientation” to higherdimensional manifolds, as a choice between two inequivalent ways in which objects can be situated with respect to their surroundings. For 2-dimensional manifolds, an orientation is essentially a choice of which rotational direction should be considered “clockwise” and which “counterclockwise.” For 3-dimensional ones, it is a choice between “left-handedness” and “right-handedness.” The general definition of an orientation is an adaptation of these everyday concepts to arbitrary dimensions. As we will see in this chapter, a vector space always has exactly two choices of orientation. In Rn , there is a standard orientation that we can all agree on; but in other vector spaces, an arbitrary choice has to be made. For manifolds, the situation is much more complicated. On a sphere, it is possible to decide unambiguously which rotational direction is counterclockwise, by looking at the surface from the outside (Fig. 15.1). On the other hand, a Möbius band (Fig. 15.2) has the curious property that a figure moving around on the surface can come back to its starting point transformed into its mirror image, so it is impossible to decide consistently which of the two possible rotational directions on the surface to call “clockwise” and which “counterclockwise,” or which is the “front” side and which is the “back” side. The analogous phenomenon in a 3-manifold would be a right-handed person who takes a long trip and comes back left-handed. Manifolds like the sphere, in which it is possible to choose a consistent orientation, are said to be orientable; those like the Möbius band in which it is not possible are said to be nonorientable. In this chapter we develop the theory of orientations of smooth manifolds. They have numerous applications, most notably in the theory of integration on manifolds, which we will study in Chapter 16. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, 377 DOI 10.1007/978-1-4419-9982-5_15, © Springer Science+Business Media New York 2013
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Fig. 15.1 A sphere is orientable
Orientations
Fig. 15.2 A Möbius band is not orientable
We begin the chapter with an introduction to orientations of vector spaces, and then show how this theory can be carried over to manifolds. Next, we explore the ways in which orientations can be induced on hypersurfaces and on boundaries of manifolds with boundary. Then we treat the special case of orientations on Riemannian manifolds and Riemannian hypersurfaces. At the end of the chapte
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