Page 45 - Curvature and Homology

1.8. BUNDLE OF FRAMES 27
affine connection, that is, the functions R$k,, are functions of the qk
only. More generally, for a tensor of type (r, s)

Now, if both the torsion and curvature tensors vanish, covariant
differentiation is symmetric. It does not follow, however, that the
qi vanish, that is, the space is not necessarily affine space.
An affinely connected manifold is said to be locally afine or locallyflat
if a coordinate system exists relative to which the coefficients of con-
nection vanish. Under the circumstances, both the torsion and curvature
tensors vanish. Conversely, if the torsion and curvature are zero it can be
shown that the manifold is locally flat (cf. 1.E).

1.8. Bundle of frames
The necessity of the concept of an affine connection on a differentiable
manifold has been clearly established from an analytical point of view.
A geometrical interpretaticn of this notion is desirable. Hence, in this
section a realization of this very important concept will be given in
terms of the bundle of frames over M.
By a frame x at the point P E M is meant a set {XI, ..., X,} of linearly
independent tangent vectors at P. Let B be the set of all frames x at all
points P of M. Every element a E GL(n, R) acts on B to the right, that
is, if a denotes the matrix (a:) and x = {XI, ..., X,}, then x a =
@{XI, ..., aiXj} E B is another frame at P. The map ir : B+ M of
B onto M defined by ~(x) = P assigns to each frame x its point of origin.
In terms of a system of local coordinates ul, ..., un in M the local
coordinates in B are given by (uj, ffiJ-the n2 functions C:,, being
defined by the n vectors Xi of the frame: