Page 117 - Curvature and Homology

3.4. INFINITESIMAL TRANSFORMATIONS 99
The infinitesimal transformation X is said to be dzgerentiable of class
k - 1 if Xf is differentiable of class k - 1 for every f of class k 2 1.
We give a geometrical interpretation of vector fields on M in terms
of groups of transformations of M which will prove particularly useful
when discussing the conformal geometry of a Riemannian manifold
as well as the local geometry of a compact semi-simple Lie group (cf.
Chapter IV). For a more detailed treatment of the results of this section
the reader is referred to [27, 631. To this end, we define a (global)
1-parameter group of diflerentiable transformations of M denoted by
9, ( - oo < t < oo) as follows:
(i) 9, is a differentiable transformation (cf. 5 1.5) of M (- oo < t < a)) ;
(ii) The map (t, P) -+ y,(P) is a differentiable map from R x M
into M;

The 1-parameter group 9, induces a (contravariant) vector field X on M
defined by the equation

-
(Xf) (P) = lim f (~t(p)) f (PI
t-r 0 t
(f: an arbitrary differentiable function) the limit being assured by
condition (ii). Under the circumstances, the vector field X is said to be
complete. On the other hand, a vector field X on M is not necessarily
induced by a global 1-parameter group 9, of M. However, associated
with a point P of M there is a neighborhood U of P and a constant
E > 0 such that for I t I < E there is a (local) 1-parameter group of
transformations F, satisfying the conditions:
(i)' 9, is a differentiable transformation of U onto ?,(U), I t I < E;
(ii)' The map (t, P) + v,(P) is a differentiable map from (- E, E) x U
into U;
(iii)' 9,+,(P) = cp,(cp,(P)), P E U provided I s I, I t I and I s + t 1
are each less than E.
Moreover, 9, induces the vector field X, that is equation (3.4.3) is
satisfied for each P E U and differentiable function f. The vector field X
is then said to generate 9,. The proof is omitted. (We shall occasionally
write cpx(P, t) for cp,(P) (cf. 1II.C)). The uniqueness of the local group 9,
is immediate. Hence the existence of a 'flow' in a neighborhood of P
is equivalent to that of a 'field of directions' at P.
If M is compact it may be shown that every vector field is complete
and in our applications this will usually be the case.