Page 114 - Curvature and Homology

% 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

more, the product of an element of Ap and one of AQ is an element
of Ap*, and this product is required to be associative.
The tensor product A @ B of the underlying spaces of the graded
algebras A and B can be made into a graded algebra by defining a suitable
multiplication and graduation in A @ B.
The exterior differential operator d is an anti-derivation in the ring
of exterior differential polynomials, that is for a p-form a and q-form /9 :

where 2; = (- 1)Pa. For an element a of Ap the involutive auto-
morphism: a + d = (- 1)Pa is called the bar operation. An endomor-
phism 8 of the additive structure of A is said to be of degree r if for
each p, 8(AP) C AP+r. As an endomorphism the operator d is of
degree + 1. An endomorphism 8 of A of even degree is called a
derivation if for any a and b of A

It is called an anti-derivation if it is of odd degree and

Evidently, if 8 is an anti-derivation, 88 is a derivation. If 8, and 8, are anti-
derivations Ole2 + O2Ol is a derivation. The bracket [el, 8J = Ole, -
of two derivations is again a derivation. Moreover, for a derivation 8,
and an anti-derivation 82, [el, 0J is an anti-derivation.
If the algebra A is generated by its elements of degrees 0 and 1, a
derivation or anti-derivation is completely determined if it is given in
A0 and A1.
Let X be an infinitesimal transformation on an n-dimensional
Riemannian manifold M. In terms of the natural bases {a/ @ul, ..a, a/ @un}
and {dul, a-0, dun) relative to the local coordinates ul, .--, un write
X = e(a/aug) and = &dug, 5 being the covariant forin of X. Now,
for any p-form a, we define the exterior product operator ~(4):

Clearly, .(a is an endomorphism of A(T*). For any (p + 1)-form
P on M,
€(S)Q A *B = (- 1)P a A 45)*/3