General manifolds

Abstract

This chapter provides the classical definitions for manifolds (via charts, which have not appeared thus far), smooth maps to and from manifolds, tangent vectors and tangent spaces and differentials of smooth maps. Special care is taken to introduce the atlas topology on manifolds and to justify topological restrictions in the definition of a manifold. It is shown explicitly that embedded submanifolds of linear spaces as detailed in earlier chapters are manifolds in the general sense. Then, sections go on to explain how the geometric concepts introduced for embedded submanifolds extend to general manifolds mostly without effort. This includes tangent bundles, vector fields, retractions, local frames, Riemannian metrics, gradients, connections, Hessians, velocity and acceleration, geodesics and Taylor expansions. One section explains why the Lie bracket of two vector fields can be interpreted as a vector field (which we omitted in Chapter 5). The chapter closes with a section about submanifolds embedded in general manifolds (rather than only in linear spaces): This is useful in preparation for the next chapter.