Page 119 - Curvature and Homology

3.5. THE DERIVATION 6(X) 101

Lemma 3.4.3. For any two infinitesimal tran.fo~mations X and Y on M,

= lim
t+O
for any f E F where tpt is the 1 -parameter group generated by X.
Associated with any f E F, there is a differentiable family of functions
g, on M such that f tp, = f + tg, where go = Xf. This follows from
lemma 3.4.2 by putting f(t, P) = f(tp,(P)) - f(P). Hence, if we set
0)t' = (tpt)* and tpt*Y= tpt*(Y)

Corollary. If qt and $I are the I-parameter groups generated by X and Y,
respectively, then [X, Y] = 0, if an only if tp, and $, commute fm every
s and t.

3.5. The derivation B(X)

We have seen that to each tangent vector field X E T on a Riemannian
manifold M there is associated an anti-derivation i(X) of degree - 1
(called the interior product by X) of the exterior algebra A(T*) of
differential forms on M. A derivation B(X) of degree 0 of the Grassman
algebra A(T) as well as A(T*) may be defined, and in fact, completely
characterized for each X E T as follows (cf. III.B.3):
(i) 8(X)d = de(X),
(ii) 6(X)f = i(X)df, f E EAT*), and
(iii) 6(X) Y = [X, Y].
Indeed, 6(X)f = i(X)df = (X, df) = @(a/ aui), (aflauj) du*) =
( af/ aui) = Xf and 6(X)df = dB(X)f = dXf; since A (T*) is gener-
ated (locally) by its homogeneous elements of degrees 0 and 1 the
derivation 6(X) may be extended to differential forms of any degree.