Differentiable Manifolds

The basic objective of the theory of differentiable manifolds is to extend the application of the concepts and results of the calculus of the ℝ ⁿ spaces to sets that do not possess the structure of a normed vector space. The differentiability of a function of ℝ ⁿ to 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.

In this chapter several additional useful concepts are introduced, which will be extensively employed in the second half of this book. It is shown that there is a one-to-one relation between vector fields on a manifold 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.

Differential forms are completely skew-symmetric tensor fields. They are applied 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. Many of the applications of differential forms are presented in subsequent chapters.

We have met the concept of integral curve of a vector field in Sect. 2.​1 and we have seen that finding such curves is equivalent to solving a system of ODEs. In this chapter we consider a generalization of this relationship defining the integral manifolds of a set of vector fields or of differential forms. We shall show that the problem of finding these manifolds is equivalent to that of solving certain systems of differential equations.

The tangent space, T _x M, to a differentiable manifold M at a point x is a vector space different from the tangent space to M at any other point y, T _y M. In general, there is no natural way of relating T _x M with T _y M if \(x \not= y\). This means that if v and w are two tangent vectors to M at two different points, e.g., v∈T _x M and w∈T _y M, there is no natural way to compare or to combine them. However, in many cases it will be possible to define the parallel transport of a tangent vector from one point to another point of the manifold along a curve. Once this concept has been defined, it will be possible to determine the directional derivatives of any vector field on M; conversely, if we know the directional derivatives of an arbitrary vector field, the parallel transport of a vector along any curve in M is determined.

In many cases, the manifolds of interest possess a metric tensor which defines an inner product between tangent vectors at each point of the manifold. Some examples are the submanifolds of an Euclidean space and the space–time, in the context of special or general relativity.

A Lie group is a group that possesses, in addition to the algebraic structure of a group, a differentiable manifold structure compatible with its algebraic structure in the sense that the group operations are differentiable functions.

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 nondegenerate 2-form, which allows us to define a Poisson bracket between real-valued functions defined on T ^∗ M. 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.