Page 131 - Curvature and Homology
P. 131
of type (0,2) where we have written O(U) for O(A)-the vector field A
being defined by duality. Clearly, the elements A of L satisfy the equation
t(u) = 0. In a coordinate neighborhood U with local coordinates
ul, -., un the tensor t(u) has the components
the divergence of which is given by
(WY)), = gik Dk(Dj a, + Di a,) + ,
2
pa),
since
The operator 6' is used in place of - 8 since t(u) is symmetric. From
the Ricci identity (1.7.19) we obtain
Now, since the tensor t(a) is symmetric and is annihilated by g, that is,
since (g,t(u)) = 0,
Integrating both sides of this relation and applying Stokes' formula we
obtain the integral formula
An application of (3 A.2) together with (3.8.3) yields:
