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About this book

This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics.

The first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations, connections, Riemannian manifolds, Lie groups, and Hamiltonian mechanics. Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics.

This second edition greatly expands upon the first by including more examples, additional exercises, and new topics, such as the moment map and fiber bundles. Detailed solutions to every exercise are also provided.

Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and a basic knowledge of analytical mechanics

Review of the first edition:

This book presents an introduction to differential geometry and the calculus on manifolds with a view on some of its applications in physics. … The present author has succeeded in writing a book which has its own flavor and its own emphasis, which makes it certainly a valuable addition to the literature on the subject. Frans Cantrijn, Mathematical Reviews

Table of Contents

Frontmatter

Chapter 1. Manifolds

Abstract
The main goal of the theory of differentiable manifolds is to extend many of the concepts and results of the multivariate calculus to sets that do not possess the structure of a normed vector space. Recall that the differentiability of a function of \(\mathbb {R}^{n}\) to \(\mathbb {R}^{m}\) means that around each interior point of its domain the function can be approximated by a linear transformation, but this requires the notions of linearity and distance, which are not present in an arbitrary set.
Gerardo F. Torres del Castillo

Chapter 2. Lie Derivatives

Abstract
In this chapter several additional basic concepts are introduced, which will be extensively employed in what follows. It is shown that, in any differentiable manifold, there is a one-to-one relation between vector fields and families of transformations of the manifold onto itself. This relation is essential in the study of various symmetries, as shown in Chaps. 4, 6 and 8, and in the relationship of a Lie group with its Lie algebra, treated in Chap. 7.
Gerardo F. Torres del Castillo

Chapter 3. Differential Forms

Abstract
Differential forms are completely antisymmetric tensor fields, and as we shall see in all the remaining chapters of this book, they constitute a very useful and versatile tool. They are employed in some areas of physics, mainly in thermodynamics and classical mechanics, and of mathematics, such as differential equations, differential geometry, Lie groups, and differential topology.
Gerardo F. Torres del Castillo

Chapter 4. Integral Manifolds

Abstract
In this chapter we show that, in a neighborhood of each point where a vector field does not vanish, one can find local coordinates where this vector field acquires its simplest expression. Just as a vector field defines a family of curves such that through each point of the manifold there passes one of these curves, it is shown that under certain conditions, a set of vector fields (or of 1-forms) defines a family of submanifolds of a fixed dimension.
Gerardo F. Torres del Castillo

Chapter 5. Connections

Abstract
In this chapter we show that by introducing a new object, called a connection, in a manifold, we are able to translate a tangent vector at a point of the manifold to another point, if these two points can be connected by means of a curve on the manifold. This is equivalent to be able to calculate directional derivatives of vector fields. As we shall see in Sects. 6.2 and 7.4, there is a connection defined in a natural manner if the manifold has a Riemannian structure or is a Lie group. The properties imposed on a connection have their origin in the study of two-dimensional surfaces in the three-dimensional Euclidean space and lead to the concept of curvature.
Gerardo F. Torres del Castillo

Chapter 6. Riemannian Manifolds

Abstract
A differentiable manifold is properly Riemannian if it is endowed with a tensor field (known as metric tensor) which defines an inner product between tangent vectors at each point of the manifold. In a properly Riemannian manifold we can define lengths, areas, etc., can relate vectors with covectors, and introduce many of the geometric concepts present in the Euclidean spaces.
Gerardo F. Torres del Castillo

Chapter 7. Lie Groups

Abstract
In many applications one encounters groups of transformations which contain one or several parameters, and on these groups we can define a structure of differentiable manifold. Two examples are the isometries of a Riemannian manifold and the symmetries of a second-order ODE, considered in Sects. 6.​1 and 4.​3, respectively. In this chapter we shall study these groups by themselves, combining their algebraic and differentiable structures. One of the main results of this combination is the fact that each of these groups has an associated Lie algebra which almost entirely determines the group.
Gerardo F. Torres del Castillo

Chapter 8. Hamiltonian Classical Mechanics

Abstract
In this chapter we start by showing that any finite-dimensional differentiable manifold M possesses an associated manifold, denoted by \(T^{*}M\), called the cotangent bundle of M, which has a naturally defined non-degenerate 2-form, which allows us to define an operation between real-valued functions defined on \(T^{*}M\), called Poisson bracket. We then apply this structure to classical mechanics and geometrical optics, emphasizing the applications of Lie groups and Riemannian geometry. Here we will have the opportunity of making use of all of the machinery introduced in the previous chapters.
Gerardo F. Torres del Castillo

Backmatter

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