Lafontaine Jacques. An Introduction to Differential Manifolds

Springer, 2015. — 408 p.Smooth manifolds are the natural generalization of curves and surfaces. The idea of a manifold appeared for the first time (and without discussion!) in 1851, in Riemann’s inaugural lecture, and allowed him to construct a satisfactory solution of the problem of analytic continuation of holomorphic functions.It took some 50 years for a precise definition to emerge. It is a question of conceptualising, not the parts of a space Rn with large n, defined by a certain number of equations, but, in a more abstract way, objects which, “ a priori ”, are not within the “ordinary” space of dimension n, for which the notion of smooth function still makes sense.There are numerous reasons to be interested in “higher” dimensions. Perhaps one of the more evident comes from classical mechanics. Describing the space of configurations of a mechanical system rapidly depends on more than three parameters: one already needs six for a solid.
The fact that it is not always desirable to consider objects as subsets of Rn is more subtle. For example, the set of directions in three-dimensional space depends on two real parameters, and naturally forms a manifold of dimension two, called the projective plane. This manifold admits numerous realizations as a subspace of Euclidean space, but these realizations are not immediately obvious and it is not clear how to select a “natural” one amongst them.
These “abstract” manifolds furnish the natural mathematical setting for classical mechanics (both configuration and phase space), but also for general relativity and particle physics.