Page 189 - Curvature and Homology

5.4. THE OPERATORS L AND /i 171

on p-forms. The operator A is therefore dual to L and lowers the degree
of a form by 2 whereas the operator L raises the degree by 2.
Moreover, if a is of bidegree (q, r), Aa is of bidegree (q - 1, r - I).
Evidently, Aa = 0 for p-forms a of degree less than 2. From (5.4.1)

where i(f) is the interior product operator, that is, the dual of the
operator 4). Following (3.3.4), we define

where a is a p-form, and, by (3.3.5)

where X is the tangent vector dual to the 1-form 6.
Since i(8,) is an anti-derivation, A52 = n. The operator A is not
a derivation. For, since a form a of bidegree (q, r) may be expressed as a
linear combination of the forms Ojl A ... A Ojq A Ok1 A ..- A OkT, and
A is linear, one need only examine the effect of A on such forms.
Indeed, since i(8,) is an anti-derivation

a similar statement holds for i(B,). Hence,

for 1 = jl = k1 and is zero for I + j,, j,, k,, -a*, k,. Thus,