Semi-Riemannian Geometry

Chapter 18 Smooth Manifolds with a Connection

In the introduction to Part III, we set out the task of developing a theory of differential geometry built upon our earlier study of curves and surfaces, but without having to involve an ambient space. Implicit in this undertaking was the aim of recovering, to the extent possible, the results presented for curves and surfaces. Chapters 14–17 have met significant parts of this agenda. Noticeably absent, however, is a discussion of “covariant derivative” and “metric”. We remedy the first of these deficits in this chapter by adding “connection” to our discussion of smooth manifolds.

18.1 Covariant Derivatives

Let M be a smooth manifold. A connection on M is a map

such that for all vector fields X, Y, Z in and all functions f in C ^∞(M) : [∇1] ∇ (X + Y, Z) =  ∇ (X, Z) +  ∇ (Y, Z).

We refer to the pair (M, ∇) as a smooth manifold with a connection. It is possible for a given smooth manifold to have more than one ...