Page 44 - Curvature and Homology

26 I. RIEMANNIAN MANIFOLDS

parallel curve through a given point tangent to a given vector. Note
that the auto-parallel curves in An are straight lines.
Affine space has the further property that functions defined in it
have symmetric second covariant derivatives. This is, however, not the
case in an arbitrary differentiable manifold. For, let f be a function
expressed in the local coordinates (ui). Then

from which

If we put
Tjt = qk - qjS

it follows that the Tjki are the components of a tensor field of type (1,2)
called the torsion tensor of the affine connection rjk. We remark at this
point, that if 6; = ck duk are a set of n2 linear differential forms in
each coordinate neighborhood defining another affine connection on M,
then it follows from the equations (1.7.4) that rtk - fjk is a tensor field.
In particular, if we put pjk = Tij, that is, if 3; = rijduk, - Gj
is a tensor field. When we come to discuss the geometry of a Riemannian
manifold we shall see that there is an affine connection whose torsion
tensor vanishes. However, even in this case, it is not true that covariant
differentiation is symmetric although for (scalar) functions this is certainly
the case. In fact, a computation shows that

where

(In the case under consideration the components Tjkz are zero). Clearly,
is
Pjk, a tensor field of type (1,3) which is skew-symmetric in its last
two indices. It is called the curvature tensor and depends only on the