Page 123 - Curvature and Homology

(ii) The point P. a = R, P, P EM depends differentiably on a E G
and P where Ra P = Ra (P).
Clearly, Re is the identity transformation of M. Hence, Ra(Ra-I(P))
= P for every a E G and P E M. The group G is said to act effectively
if Ra P = P for every P E M implies a = e.
Let A be an element of the Lie algebra L of G and a, the 1-parameter
subgroup of G generated by A. A is a left invariant infinitesimal trans-
formation of G. The corresponding 1-parameter group of trans-
formations Rat on M induces a differentiable vector field A* on M.
Let a denote the map sending A EL to A* EL* (the Lie algebra of
differentiable vector fields on M).

Lemma 3.6.1. The map a : L -+ L* is a homomorphism.
Indeed, for any P E M denote by a, the map from G to M defined
by U~X) x. Then
= P

where is the induced map in T, (the tangent space at e E G).
Clearly, a is linear. For any two elements A and B of L, set A* = o(A)
and B* = o(B). Then, from lemma 3.4.3

B* - R,; B*
[A*, B*] = lim
t-ro t

u+B, - ~p.ad(a;-~)B,
[A*, B*Ip = lim
t+O t
B, - ad(ayl) B,
= a+ lim
t-ro t

If G acts effectively on M, a is an isomorphism. Indeed, if 4A) = 0
for some A EL, the associated 1-parameter subgroup Rag is trivial.
Since G is effective we have a, = e, from which A = 0.
We remark that the derivations O(A*) correspond to the action of
G on M.