- FIRST-ORDER DIFFERENTIAL OPERATORS ON FORMS 283
where { ei} ~=l is a (local) orthonormal frame field. If Mn is compact, the
formal adjoint of J with respect to the L^2 inner product
induced by the metric g is denoted by
J : C^00 (T Mn) -t C^00 (T Mn ©s T Mn).
By integrating by parts, it is easy to verify that
(J*e) = ~ce~gfor all covector fields e, where the Lie derivative £ and metric dual eH are
defined above. Notice that this formula can be used to define J* even when
Mn is not compact.
- First-order differential operators on forms
A p-form is a smooth section of the bundle f\P (T* Mn), namely an ele-
ment of
OP (Mn)~ C^00 (/\P (T* Mn)).
The exterior derivative is the family of operators
d = dp: OP (Mn) -7 op+l (Mn)defined for all p-forms e and vector fields Y 1 , ... , Yp by
de(Yo,Y1, ... ,Yp)~ 'L:o::;i::;p (-l)i e (Yo, Y1,... , ii, ... , Yp)
- 'L:o::;i<j:Sp (-l)i+j e ([Yi, }j], Yo, Y1, ... , ii, ... , YJ, ... , Yp) ,
where the symbol ii means that term is omitted. Although independent of
any Riemannian metric g on Mn, the exterior derivative may be expressed
with respect to the covariant derivative \7 defined by the Levi-Civita con-
nection of that metric byde (Yo, Y1, ... 'Yp) = I:f=o (-l)i (\i'Yie) (Yo, Y1, ... 'ii, ... ' Yp).
If Mn is compact, the formal adjoint of d with respect to the L^2 inner
product (-, ·) induced by the metric g is the family of operatorsJ = Sp+l : op+l (Mn) -7 OP (Mn)
defined by the requirement that(de, TJ) = (e, Jry)