Titlebook: Introduction to Smooth Manifolds; John M. Lee Textbook 20031st edition Springer Science+Business Media New York 2003 Algebra.Cohomology.De

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Textbook 20031st editionding "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, com­ puter

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Lie Group Actions,uently turns out to be a Lie group acting smoothly on .. The properties of the group action can shed considerable light on the properties of the structure. This chapter is devoted to studying Lie group actions on manifolds.

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Differential Forms,old theory, through two applications. First, as we will see in Chapter 14, they are the objects that can be integrated in a coordinate-independent way over manifolds or submanifolds; second, as we will see in Chapter 15, they provide a link between analysis and topology by way of de Rham cohomology.

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0072-5285 nder­ standing "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, c

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Vector Bundles, this kind of structure arises quite frequently—a collection of vector spaces, one for each point in ., glued together in a way that looks . like the Cartesian product of . with ℝ., but globally may be “twisted.” Such a structure is called a vector bundle.

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Tensors,ns in a coordinate-independent way. In this chapter we carry this idea much further, by generalizing from linear objects to multilinear ones. This leads to the concepts of tensors and tensor fields on manifolds.

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Orientations, by the vectors. These signs will cause problems, however, when we try to integrate differential forms on manifolds, for the simple reason that the transformation law for an .-form under a change of coordinates involves the determinant of the Jacobian, while the change of variables formula for multiple integrals involves the . of the determinant.

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De Rham Cohomology, of the manifold, connected with the existence of “holes” of higher dimensions. Making this dependence quantitative leads to a new set of invariants of smooth manifolds, called the de Rham cohomology groups, which are the subject of this chapter.