Smooth Manifolds
In this chapter, we begin by introducing the simplest type of manifolds, the topological manifolds, which are topological spaces with three special properties that encode what we mean when we say that they “locally look like ℝ n .” We then prove some impo
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Smooth Manifolds
This book is about smooth manifolds. In the simplest terms, these are spaces that locally look like some Euclidean space Rn , and on which one can do calculus. The most familiar examples, aside from Euclidean spaces themselves, are smooth plane curves such as circles and parabolas, and smooth surfaces such as spheres, tori, paraboloids, ellipsoids, and hyperboloids. Higher-dimensional examples include the set of points in RnC1 at a constant distance from the origin (an n-sphere) and graphs of smooth maps between Euclidean spaces. The simplest manifolds are the topological manifolds, which are topological spaces with certain properties that encode what we mean when we say that they “locally look like” Rn . Such spaces are studied intensively by topologists. However, many (perhaps most) important applications of manifolds involve calculus. For example, applications of manifold theory to geometry involve such properties as volume and curvature. Typically, volumes are computed by integration, and curvatures are computed by differentiation, so to extend these ideas to manifolds would require some means of making sense of integration and differentiation on a manifold. Applications to classical mechanics involve solving systems of ordinary differential equations on manifolds, and the applications to general relativity (the theory of gravitation) involve solving a system of partial differential equations. The first requirement for transferring the ideas of calculus to manifolds is some notion of “smoothness.” For the simple examples of manifolds we described above, all of which are subsets of Euclidean spaces, it is fairly easy to describe the meaning of smoothness on an intuitive level. For example, we might want to call a curve “smooth” if it has a tangent line that varies continuously from point to point, and similarly a “smooth surface” should be one that has a tangent plane that varies continuously. But for more sophisticated applications it is an undue restriction to require smooth manifolds to be subsets of some ambient Euclidean space. The ambient coordinates and the vector space structure of Rn are superfluous data that often have nothing to do with the problem at hand. It is a tremendous advantage to be able to work with manifolds as abstract topological spaces, without the excess baggage of such an ambient space. For example, in general relativity, spacetime is modeled as a 4-dimensional smooth manifold that carries a certain geometric structure, called a J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, DOI 10.1007/978-1-4419-9982-5_1, © Springer Science+Business Media New York 2013
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1 Smooth Manifolds
Fig. 1.1 A homeomorphism from a circle to a square
Lorentz metric, whose curvature results in gravitational phenomena. In such a model there is no physical meaning that can be assigned to any higher-dimensional ambient space in which the manifold lives, and including such a space in the model would complicate it needlessly. For such reasons, we need to think
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