Page 120 - Curvature and Homology

1 02 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
On the other hand, by conditions (ii) and (iii), 8(X) may be extended to
all of A(T). In fact, O(X) may be extended to the tensor algebras of
contravariant and covariant tensors by insisting that (for each X) it be a
derivation of these algebras. For example, by lemma 3.4.3
Y - 9t.Y
4X)Y = lim
t* t
where q~, is the 1-parameter group generated by X. Hence, for any
tensor t of type (p, 0)
t - t
9:.
B(X) t = lim
s-r6 S
where q~:, = @ @ vs. (p times) is the induced map in T:. (For
any XI, -, X, E T, &(XI @ *.. @ Xp) = (pl.(X1) @ @ va.(Xg)).
Since [8(X), 8(Y)] = 8(X) 8(Y) - B(Y) 8(X) is a derivation, it follows
from the Jacobi identity that the map x L.B(X) is a representation
of the Lie algebra of tangent vector fields.

Lemma 3.5.1. The derivations d, i(X), and B(X) are dated by the
formula
B(X) = i(X)d + di(X). (3.5.1)

Since both sides are derivations, and since the Grassman algebra of
differential forms is generated by its homogeneous elements of degrees
0 and 1, the relation need only be established for differential forms of
degrees 0 and 1 :

Lemma 3.5.2. For a I-form u on M and any tangent vector jieldr X
and Y on M :
<X A Y, da) = Xar(Y) - Yar(X) - a([X,Y]). (3.5.2)
The right hand side is meaningful since at each point P of M, Tp
and T,* are dual vector spaces. Thus, u is a linear map from T into F.
By linearity, it is sufficient to prove the relation for X = a/ihi,
Y = a/au* and u = gdf where f and g are functions expressed in the
coordinates (3). In fact, if the relation holds for a, j3 E A1(T*), it holds
for u + /3 and fa where f is a differentiable function. We may therefore
assume u = duk and in this case, both sides vanish.