Page 155 - Curvature and Homology

4.3. LOCAL GEOMETRY 137
In terms of a system of local coordinates ul, ..., un the vector fields
X& = 1, ..., n) may be expressed as Xa = fi(a/aui). Since G is
completely parallelisable, the n x n matrix (6;) has rank n, and so, if
we put
g" = ft f; (4.3.3)
the matrix (gij) is positive definite and symmetric. We may therefore
define a metric g on G by means of the quadratic form

where thegjk are elements of the matrix inverse to (gjk). Again, the metric
tensor g may be used to raise and lower indices in the usual manner.
It should be remarked that the metric is completely determined by
the group G.
We now define n covariant vector fields va(a = 1, ..a, n) on G with
components f?(i = 1, .-., n) (relative to the given system of local
coordinates) by the formulae
c = gap 4 gu.
It follows easily that

However, it does not follow that, in the metric g the X,(a = 1, ..a, n) are
orthonormal vectors at each point of G.
A set of n2 linear differential forms oj = c&uk is introduced in
each coordinate neighborhood by putting

By virtue of the equations (4.3.6) the qk may be written as

It is easily verified that equations (1.7.3) are satisfied in the overlap
of two coordinate neighborhoods. The d forms oj in each coordinate
neighborhood define therefore an affine connection on G. The torsion
tensor Tjkf of this connection may be written as

(The factor is introduced for reasons of convenience (cf. 1.7.18)).