Page 41 - Curvature and Homology

1.7. AFFINE CONNECTIONS 23

1.7. Affine connections
We have seen that the partial derivatives of a function with respect to a
given system of local coordinates are the components of a covariant
vector field or, stated in an invariant manner, the differential of a function
is a covector. That this case is unique has already been shown (cf.
equation 1.3.10). A similar computation for the contravariant vector
field X = ti(a/ ad) results in

where

in U n t'?. Again, the presence of the second term on the right indicates
that the derivative of a contravariant vector field does not have tensor
character. Differentiation may be given an invariant meaning on a
manifold by introducing a set of n2 linear differential forms wj = qk duk
in each coordinate neighborhood, so that in the overlap U n 7? of
two coordinate neighborhoods

A direct computation shows that in the intersection of three coordinate
neighborhoods one of the relations (1.7.3) is a consequence of the
others. In terms of the n3 coefficients qk, equations (1.7.3) may be
written in the form

These equations are the classical equations of transformation of an
affine connection. With these preliminaries we arrive at the notion
we are seeking. We shall see that the wj permit us to define an invariant
type of differentiation over a differentiable manifold.
An afine connection on a differentiable manifold M is defined by
prescribing a set of n2 linear differential forms wf in each coordinate
neighborhood of M in such a way that in the overlap of two coordinate
neighborhoods