Page 121 - Curvature and Homology

Equation (3.5.2) indicates a (local) relationship between the derivations
B(X) and d. Indeed, if we write
1
dd =-cikCUj A &Jk, cifk f cij =O (3.5.3)
2
and
@(Xj) x, = - bif, X*, b:, + b:, = 0, (3.5.4)
where {X*} and {uk) are dual bases, then

from which by (1.5.1), (3.5.3) and (3.5.4)

The reader is referred to Chapter IV where this relationship is exploited
more fully. We remark that equation (3.5.2) has important implications
in the theory of connections as well [63].

3.6. Lie transformation groups [TI, 631
A Lie poup G is a group which is simultaneously a differentiable
manifold (the points of the manifold coinciding with the elements of the
group) in which the group operation (a, b) -+ ab-I (a, b E G) is a
differentiable map of G x G into G. It is well-known that as a manifold
G admits an analytic structure in such a way that the group operations
in G are analytic. It follows that the map x -+ ax is analytic. We denote
this map by La and call it the left translation in G by a. Hence, every
left translation La is an analytic homeomorphism of G (as an analytic
manifold) with itself. It follows that if x and y are any two elements
of G, there exists an element a = yx-I such that the induced map
La. = (La), maps T, isomorphically onto T,.
An infinitesimal transformation X on G is said to be left invariant
if for every a€ G, La ,Xe = X,. Hence, associated with an element AE Te,
where e E G is the identity, there is a unique left invariant infini-
tesimal transformation X which takes the value A at e. It can be shown
that every left invariant infinitesimal transformation is analytic. Let fi
denote the set of left invariant infinitesimal transformations of G;
L is a vector space over R of dimension equal to that of G. In fact, if
to a tangent vector Xe E Te we associate the infinitesimal transformation
X EL defined by Xu = La,Xe (a E G) it is seen that as vector spaces