The one major item of discussion from Chapters 1 to 9 which has not been generalised so as to apply in a differentiable manifold is the idea of a connection, that is, of parallelism and the associated operation of covariant differentiation. This is the subject of the present chapter.

It may be recalled that in an affine space it makes sense to say whether two vectors defined at different points are parallel, because they may be compared with the help of the natural identification of tangent spaces at different points. On a surface, on the other hand, no such straightforward comparison of tangent vectors at different points is possible; there is however a plausible and workable generalisation from affine spaces to surfaces in which the criterion of parallelism depends on a choice of path between the points where the tangent vectors are located. Though the covariant differentiation operator associated with this path-dependent parallelism satisfies the first order commutation relation of affine covariant differentiation, ∇UV – ∇VU – [U,V] – 0, it fails to satisfy the second order one, ∇U∇VW – ∇V∇UW – ∇[U,V]W – 0 in general; and indeed its failure to do so is intimately related to the curvature of the surface.

In generalising these notions further, from surfaces to arbitrary differentiable manifolds, we have to allow for the arbitrariness of dimension; we have to develop the theory without assuming the existence of a metric in the first instance (though we shall consider that important specialisation in due course); and we have to allow for the possibility that not even the first order commutation relation survives.