Page 42 - Curvature and Homology

24 I. RIEMANNIAN MANIFOLDS

A manifold with an affine connection is called an aflnely connected
manqold.
The existence of an affine connection on a differentiable manifold
will be shown in 5 1.9. In the sequel, we shall assume that M is an
affinely connected manifold. Now, from the equations of transformation
of a contravariant vector field X = fC(a/aui) we obtain by virtue of the
equations (1.7.5)
dfi = dp: + P: dfj
(1.7.6)
= (w: - 6: P;) fj + +j dp.
By rewriting these equations in the symmetrical form

we see that the quantity in brackets transforms like a contravariant
vector field. We call this quantity the covariant dzflmential of X and
denote it by DX Its jth component dp + oi fk will be denoted by
(DW. In terms of the natural base for covectors, (1.7.7) becomes

ay + 8";) duj = p;(w aP' + p rg) :)dug.
(33
We set

and call it the covariant derivative of X with respect to u'. That the
components%, (J transform like a tensor field of type (1,l) is clear.
In fact, it follows from (1.7.8) that

where the 1.h.s. denotes the covariant derivative of X with respect to z2.
A similar discussion in the case of the covariant vector field ti permits
us to define the covariant derivative of fi as the tensor field Dj& of
type (0,2) where
2;
D, Ei = - - tk r;. (1.7.1 1)

The extension of the above argument to tensor fields of type (r, s) is
straightforward-the covariant derivative of the tensor field @...irjl...j,
with respect to uk being