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2012 | Buch

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Introduction to Smooth Manifolds

verfasst von: John M. Lee

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.

This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.

Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Smooth Manifolds

Abstract

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 ℝⁿ.” We then prove some important topological properties of manifolds that we use throughout the book. In the second section we introduce an additional structure, called a smooth structure, that can be added to a topological manifold to enable us to do calculus. Following the basic definitions, we introduce a number of examples of manifolds, so you can have something concrete in mind as you read the general theory. At the end of the chapter we introduce the concept of a smooth manifold with boundary, an important generalization of smooth manifolds that will have numerous applications throughout the book.

John M. Lee

Chapter 2. Smooth Maps

Abstract

The main reason for introducing smooth structures was to enable us to define smooth functions on manifolds and smooth maps between manifolds. In this chapter we carry out that project. We begin by defining smooth real-valued and vector-valued functions, and then generalize this to smooth maps between manifolds. We then focus our attention for a while on the special case of diffeomorphisms, which are bijective smooth maps with smooth inverses. If there is a diffeomorphism between two smooth manifolds, we say that they are diffeomorphic. The main objects of study in smooth manifold theory are properties that are invariant under diffeomorphisms. At the end of the chapter, we introduce a powerful tool for blending together locally defined smooth objects, called partitions of unity. They are used throughout smooth manifold theory for building global smooth objects out of local ones.

John M. Lee

Chapter 3. Tangent Vectors

Abstract

The central idea of calculus is linear approximation. In order to make sense of calculus on manifolds, we need to introduce the tangent space to a manifold at a point, which we can think of as a sort of “linear model” for the manifold near the point. Motivated by the fact that vectors in ℝⁿ act on smooth functions by taking their directional derivatives, we define a tangent vector to a smooth manifold to be a linear map from the space of smooth functions on the manifold to ℝ that satisfies a certain product rule. After defining tangent vectors, we show how a smooth map between manifolds yields a linear map between tangent spaces, called the differential of the map, and a smooth curve determines a tangent vector at each point, called its velocity. In the final two sections we discuss and compare several other approaches to defining tangent spaces, and give a brief overview of the terminology of category theory, which puts the tangent space and differentials in a larger context.

John M. Lee

Chapter 4. Submersions, Immersions, and Embeddings

Abstract

In this chapter we study three classes of smooth maps whose local behavior is accurately modeled by the behavior of their differentials: smooth submersions (whose differentials are surjective everywhere), smooth immersions (whose differentials are injective everywhere), and smooth embeddings (injective smooth immersions that are also homeomorphisms onto their images). Smooth immersions and embeddings, as we will see in the next chapter, are essential ingredients in the theory of submanifolds, while smooth submersions play a role in smooth manifold theory closely analogous to the role played by quotient maps in topology. The engine that powers this discussion is the rank theorem, a corollary of the inverse function theorem, which we prove in the first section of the chapter. Then we delve more deeply into smooth embeddings and smooth submersions, and apply the theory to a particularly useful class of smooth submersions, the smooth covering maps.

John M. Lee

Chapter 5. Submanifolds

Abstract

Many familiar manifolds appear naturally as smooth submanifolds, which are smooth manifolds that are subsets of other smooth manifolds. As you will soon discover, the situation is quite a bit more subtle than the analogous theory of topological subspaces. We begin by defining the most important type of smooth submanifolds, called embedded submanifolds, which have the subspace topology inherited from their containing manifolds. Next, we introduce a more general kind of submanifolds, called immersed submanifolds, which turn out to be the images of injective immersions. At the end of the chapter, we show how the theory of submanifolds can be generalized to the case of submanifolds with boundary.

John M. Lee

Chapter 6. Sard’s Theorem

Abstract

This chapter introduces a powerful tool in smooth manifold theory, Sard’s theorem, which says that the set of critical values of a smooth function has measure zero. After proving the theorem, we use it to prove three important results about smooth manifolds. The first result is the Whitney embedding theorem, which says that every smooth manifold can be smoothly embedded in some Euclidean space. (This justifies our habit of visualizing manifolds as subsets of ℝⁿ.) The second result is the Whitney approximation theorem, which comes in two versions: every continuous real-valued or vector-valued function can be uniformly approximated by smooth ones, and every continuous map between smooth manifolds is homotopic to a smooth map. The third result is the transversality homotopy theorem, which says, among other things, that embedded submanifolds can always be deformed slightly so that they intersect “nicely” in a certain sense that we will make precise.

John M. Lee

Chapter 7. Lie Groups

Abstract

In this chapter we introduce Lie groups, which are smooth manifolds that are also groups in which multiplication and inversion are smooth maps. Besides providing many examples of interesting manifolds themselves, they are essential tools in the study of more general manifolds, primarily because of the role they play as groups of symmetries of other manifolds. We begin with the definition of Lie groups and some of the basic structures associated with them, and then present a number of examples. Next we study Lie group homomorphisms, which are group homomorphisms that are also smooth maps. Then we introduce Lie subgroups (subgroups that are also smooth submanifolds), which lead to a number of new examples of Lie groups. After explaining these basic ideas, we introduce actions of Lie groups on manifolds, which are the primary raison d’être of Lie groups. At the end of the chapter, we briefly touch on group representations.

John M. Lee

Chapter 8. Vector Fields

Abstract

Vector fields are familiar objects of study in multivariable calculus. In this chapter we show how to define vector fields on smooth manifolds, as certain kinds of maps from the manifold to its tangent bundle. Then we introduce the Lie bracket operation, which is a way of combining two smooth vector fields to obtain another. The most important application of Lie brackets is to Lie groups: the set of all smooth vector fields on a Lie group that are invariant under left multiplication is closed under Lie brackets, and thus forms an algebraic object naturally associated with the group, called the Lie algebra of the Lie group. We show how Lie group homomorphisms induce homomorphisms of their Lie algebras, from which it follows that isomorphic Lie groups have isomorphic Lie algebras.

John M. Lee

Chapter 9. Integral Curves and Flows

Abstract

The primary geometric objects associated with smooth vector fields are their integral curves, which are smooth curves whose velocity at each point is equal to the value of the vector field there. The collection of all integral curves of a given vector field on a manifold determines a family of diffeomorphisms of (open subsets of) the manifold, called a flow. The main theorem of this chapter, the fundamental theorem on flows, asserts that every smooth vector field determines a unique maximal integral curve starting at each point, and the collection of all such integral curves determines a unique maximal flow. After proving the fundamental theorem, we show how “flowing out” from initial submanifolds along vector fields can be used to create useful parametrizations of larger submanifolds. We then introduce the Lie derivative, which is a coordinate-independent way of computing the rate of change of one vector field along the flow of another. In the last section, we apply flows to the study of first-order partial differential equations.

John M. Lee

Chapter 10. Vector Bundles

Abstract

In this chapter, we introduce an important generalization of tangent bundles: if M is a smooth manifold, a vector bundle over M is a collection of vector spaces, one for each point in M, glued together to form a manifold that looks locally like the Cartesian product of M with ℝⁿ, but globally may be “twisted.” We then go on to discuss local and global sections of vector bundles (which correspond to vector fields in the case of the tangent bundle). At the end of the chapter, we discuss the natural maps between bundles, called bundle homomorphisms, and subsets of vector bundles that are themselves vector bundles, called subbundles.

John M. Lee

Chapter 11. The Cotangent Bundle

Abstract

In this chapter we introduce a construction that is not typically seen in elementary calculus: tangent covectors, which are linear functionals on a tangent space to a smooth manifold M. The space of all covectors at p∈M is a vector space called the cotangent space at p; in linear-algebraic terms, it is the dual space to T _p M. The union of all cotangent spaces at all points of M is a vector bundle called the cotangent bundle. Whereas tangent vectors give us a coordinate-free interpretation of derivatives of curves, it turns out that derivatives of real-valued functions on a manifold are most naturally interpreted as tangent covectors. Thus we define the differential of a real-valued function as a covector field (a smooth section of the cotangent bundle); it is a coordinate-independent analogue of the gradient. In the second half of the chapter we introduce line integrals of covector fields, which satisfy a far-reaching generalization of the fundamental theorem of calculus.

John M. Lee

Chapter 12. Tensors

Abstract

Much of the technology of smooth manifold theory is designed to allow the concepts of linear algebra to be applied to smooth manifolds. Calculus tells us how to approximate smooth objects by linear ones, and the abstract definitions of manifold theory give a way to interpret these linear approximations in a coordinate-independent way. In this chapter we carry this idea much further, by generalizing from linear maps to multilinear ones—those that take several vectors as input and depend linearly on each one separately. This leads to the concepts of tensors and tensor fields on manifolds. Tensors will pervade the rest of the book, and we will see significant applications of them when we study Riemannian metrics, differential forms, orientations, integration, de Rham cohomology, foliations, and symplectic structures.

John M. Lee

Chapter 13. Riemannian Metrics

Abstract

In this chapter, for the first time, we introduce geometry into smooth manifold theory. To define geometric concepts such as lengths and angles on a smooth manifold, we introduce a structure called a Riemannian metric, which is a choice of inner product on each tangent space, varying smoothly from point to point. After defining Riemannian metrics and the main constructions associated with them, we show how submanifolds of Riemannian manifolds inherit induced Riemannian metrics. Then we show how a Riemannian metric leads to a distance function, which allows us to consider connected Riemannian manifolds as metric spaces.

John M. Lee

Chapter 14. Differential Forms

Abstract

In this chapter, we begin to develop the theory of differential forms, which are alternating tensor fields on manifolds. It might come as a surprise, but these innocent-sounding objects turn out to be considerably more important in smooth manifold theory than symmetric tensor fields, because they provide a framework for generalizing a variety of concepts from multivariable calculus. The purpose of this chapter is to describe the main tools for manipulating differential forms. The most important such tool is the exterior derivative, which generalizes the differential of a smooth function that we introduced in Chapter 11, as well as the gradient, divergence, and curl operators of multivariable calculus. At the end of the chapter, we will see how the exterior derivative can be used to simplify the computation of Lie derivatives of differential forms.

John M. Lee

Chapter 15. 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.

John M. Lee

Chapter 16. Integration on Manifolds

Abstract

In Chapter 11, we introduced line integrals of covector fields, which generalize ordinary integrals to the setting of curves in manifolds. It is also useful to generalize multiple integrals to manifolds. In this chapter, we carry out that generalization. As we show in the beginning of this chapter, there is no way to define the integral of a function in a coordinate-independent way on a smooth manifold. On the other hand, differential forms turn out to have just the right properties for defining integrals intrinsically. After defining integrals of differential forms over oriented smooth manifolds, we prove one of the most important theorems in differential geometry: Stokes’s theorem. It is a generalization of the fundamental theorem of calculus, the fundamental theorem for line integrals, and the classical theorems of vector analysis. Next, we show how these ideas play out on a Riemannian manifold. At the end of the chapter, we introduce densities, which are fields that can be integrated on any manifold, not just oriented ones.

John M. Lee

Chapter 17. De Rham Cohomology

Abstract

In Chapter 14, we defined closed and exact differential forms, and showed that every exact form is closed. In this chapter, we explore the converse question: Is every closed form exact? The answer is locally yes, but globally no. The question of which closed forms are exact depends on subtle topological properties of the manifold, connected with the existence of “holes” of various 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. They are easily shown to be diffeomorphism invariants, but surprisingly they turn out also to be topological invariants. We prove a general theorem, called the Mayer–Vietoris theorem, that expresses the de Rham groups of a manifold in terms of those of its open subsets. Using it, we compute the de Rham groups of spheres and the top-degree groups of compact manifolds, and give a brief introduction to degree theory for maps between compact manifolds of the same dimension.

John M. Lee

Chapter 18. The de Rham Theorem

Abstract

The topological invariance of the de Rham groups suggests that there should be some purely topological way of computing them. There is indeed: in this chapter, we prove the de Rham theorem, which says that the de Rham groups of a smooth manifold are naturally isomorphic to its singular cohomology groups, which are algebraic-topological invariants that measure “holes” in a rather direct sense. First we give a very brief introduction to singular homology and cohomology, and prove that they can be computed by restricting attention only to smooth simplices. In the final section of the chapter, we prove the de Rham theorem by showing that integration of differential forms over smooth simplices induces isomorphisms between the de Rham groups and the singular cohomology groups.

John M. Lee

Chapter 19. Distributions and Foliations

Abstract

Given a nonvanishing vector field on a smooth manifold M, the results of Chapter 9 show that the integral curves of the vector field fill up M and fit together nicely like parallel lines in Euclidean space. In this chapter we explore an important generalization of this idea to higher-dimensional submanifolds. Given a smooth subbundle of the tangent bundle of M, called a distribution on M, we can ask whether there are k-dimensional submanifolds (called integral manifolds of the distribution) whose tangent spaces at each point are the given subspaces of the tangent bundle. The answer in this case is more complicated than in the case of vector fields: there is a nontrivial necessary condition, called involutivity, that must be satisfied by the distribution. The main theorem of this chapter, the Frobenius theorem, tells us that involutivity is also sufficient for the existence of an integral manifold through each point. At the end of the chapter, we give applications of the theory to Lie groups and to partial differential equations.

John M. Lee

Chapter 20. The Exponential Map

Abstract

In this chapter we introduce the exponential map of a Lie group, which is a canonical smooth map from the Lie algebra into the group, mapping lines through the origin in the Lie algebra to one-parameter subgroups. As our first application, we prove the closed subgroup theorem, which says that every topologically closed subgroup of a Lie group is actually an embedded Lie subgroup. Next we prove a higher-dimensional generalization of the fundamental theorem on flows: if G is a simply connected Lie group, then any Lie algebra homomorphism from its Lie algebra into the set of complete vector fields on a smooth manifold M generates a smooth action of G on M. Using this theorem, we prove that there is a one-to-one correspondence between isomorphism classes of finite-dimensional Lie algebras and isomorphism classes of simply connected Lie groups. At the end of the chapter, we show that connected normal subgroups of a Lie group correspond to ideals in its Lie algebra.

John M. Lee

Chapter 21. Quotient Manifolds

Abstract

In Chapter 4, we studied surjective smooth submersions, and emphasized their relationships with quotient maps. But one question we did not address there was which quotients of smooth manifolds are themselves smooth manifolds. The quotients that are easiest to analyze are those resulting from smooth Lie group actions. The main theorem of the chapter is the quotient manifold theorem, which asserts that a smooth Lie group action yields a quotient space with a natural smooth manifold structure provided that it is free (meaning that the group acts without fixed points) and proper (which means roughly that each compact subset is moved away from itself by most elements of the group). After proving the theorem, we apply it to the study of actions by discrete groups, which under suitable conditions yield covering maps, and homogeneous spaces, which are smooth manifolds endowed with smooth transitive Lie group actions. At the end of the chapter we describe a number of applications to the theory of Lie groups.

John M. Lee

Chapter 22. Symplectic Manifolds

Abstract

In this final chapter we introduce a new kind of geometric structure on manifolds, called a symplectic structure, which is superficially similar to a Riemannian metric but turns out to have profoundly different properties. It is simply a choice of a closed, nondegenerate 2-form. Symplectic structures have surprisingly varied applications in mathematics and physics, including partial differential equations, differential topology, and classical mechanics, among many other fields. After defining symplectic structures, we give a proof of the important Darboux theorem, which shows that every symplectic form can be put into canonical form locally by a choice of smooth coordinates. Then we give a brief introduction to Hamiltonian systems, which are central to the study of classical mechanics, and to an odd-dimensional analogue of symplectic structures, called contact structures. At the end of the chapter, we show how symplectic and contact geometry can be used to construct solutions to first-order partial differential equations.

John M. Lee

Backmatter

Titel: Introduction to Smooth Manifolds
verfasst von: John M. Lee
Verlag: Springer New York
Electronic ISBN: 978-1-4419-9982-5
Print ISBN: 978-1-4419-9981-8
DOI: https://doi.org/10.1007/978-1-4419-9982-5