Page 47 - Curvature and Homology

As an exercise we leave to the reader the verification of the formulae
for the exterior derivatives of the 0j and 0{:

dei - e: A e: = @,
where

and

-the P,,,? and Sj,, being functions on B whereas the torsion and
curvature tensors are defined in M. Equations (1.8.7) - (1.8.9) are called
the equations of structure. They are independent of the particular choice
of frames, so that if we consider only those frames for which

and
dwi - w: A w{ = - #pi, du' A durn. (1.8.12)
Differentiating equations (1.8.7) and (1.8.8) we obtain the Bianchi
identities:

We have seen that an affine connection on M gives rise to a complete
parallelisability of the bundle of frames B over M, that is the affine
connection determines n2 + n linearly independent linear differential
forms in B. Conversely, if nd linear differential forms 0: are given in B
which together with the n-forms 81 satisfy the equations of structute,
they define an affine connection. The proof of this important fact is
omitted.
Let a be an element of the structural group GL(n, R) of the bundle
of frames B over M. It induces a linear isomorphism of the tangent