Page 88 - Curvature and Homology

70 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

Then
dujl A ... A du**-P (2.7.8)
*a = a*(jl...jn-p)
where
a* - a(il...ip).
jl.. -j,,-# ' (il.. .ip)jl.. . j,+p (2.7.9)
In the last sum, only the terms corresponding to the values of i,, *--, $
which are different from j,, ..a, j,, can be non-zero. The form *a 1s
called the adjoint of the form a. That the form (2.7.6) is the adjoint of
the form df = (af/aui) dui is an easy exercise. The adjoint of the (constant)
function 1 (considered as a form of degree 0) is the volume element

The adjoint of any function, considered as a 0-form, is its product with
the volume element.
If A and B are vectors in E3 with,the natural orientation, and the
* operation is defined in terms of the natural Riemannian structure of ES,
then * (A A B) is usually called the vector product of A and B. In E2,
the * operator applied to vectors is essentially the operation of a rotation
through 42 radians.
As in 5 2.6 the operator * has the-properties:
(i) *(a + /3) = *a + */3, *dfor) =f(*a),
(ii) **a = *(*a) = (- l)W+Pa,
(iii) a A */3 = /3 A *a,
(iv) a A *a = 0, if and only if, a = 0 where a and /3 are forms of
degree p and f is a 0-form (function).
Let
OL = a(il...ip) duil A ... A duiv,
and
p, duil A ... A dui.;
B = b (il

The proof of property (ii) and (2.7.11) follows by choosing an ortho-
normal coordinate system at a point. Hence, the relation between a
and *a is symmetrical, save perhaps for sign.
We define the (global) scalar product (a, /3) of a and /3 as the (real)
number