The treatment of surfaces in Chapter 9 shows that restricting oneself to affine spaces and subspaces would be a limitation on geometrical thinking much too severe to be tolerable. The more general idea foreshadowed there and developed in the rest of this book is that of a manifold, a space which locally resembles an affine space but globally may be quite different. Manifolds are used in Lagrangian and Hamiltonian mechanics, where configuration space and phase space are among the relevant examples, in general relativity theory, where space-time is a manifold and no longer an affine space, and in the theory of groups: the rotation and Lorentz groups considered in Chapter 8 are among those which may to advantage be considered to be manifolds.
In this chapter we define manifolds and maps between them, and go on to explain what modifications must be made to the ideas introduced, in the affine space context, in Chapters 1 to 7 to adapt these ideas to manifolds.
Manifolds Defined
We begin with two examples. Like the sphere, dealt with in the previous chapter, these examples lack the property possessed by affine spaces that one may label points with a single coordinate system so that
(1) nearby points have nearby coordinates, and
(2) every point has unique coordinates.
On the other hand, as in the case of the sphere, it is possible to choose for each of these examples a set of coordinate systems such that
(1) nearby points have nearby coordinates in at least one coordinate system,
and
(2) every point has unique coordinates in each system which covers it.