Page 126 - Curvature and Homology

108 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
Assume that the vector field X defines an infinitesimal motion on M.
Then, 8(X)g vanishes, that is
Dj ti + Di fj = O*
It follows that

Hence, applying the Bianchi identity (1.10.24) and the interchange
formula (1.7.19) for covaria'nt derivatives

We conclude that

(This means that the Lie derivative of the affine connection vanishes
or, what is the same, 8(X) commutes with the operator of covariant
differentiation (cf. 5 3.10)). On the other hand, if X is a solution
of these equations it need not be an infinitesimal motion (cf. 5 3.10).
In the case where M is En, if we choose a cartesian coordinate system
(xl, ..., xn) equations (3.7.5) and (3.7.6) reduce to
@ ti
a fi
-+-=O and --
axj axi ax, a* - O-
Integrating, we obtain

The vector whose components are the a, is the translation part of the
motion whereas the tensor with components aij defines a rotation about
the origin.
The infinitesimal motion X is usually called a Killing vector fild.
Let L be a subalgebra of- the Lie algebra T of tangent vector fields
on M. A p-form on M is said to be L-invariant if it is a zero of all the
derivations 8(X) for X E L. Clearly, the L-invariant differential forms
constitute a subalgebra of the Grassman algebra of differential forms
on M. Moreover, this subalgebra is stable under the operator d. This
follows from property (i) of 5 3.5.
Let a and fi be any two p-forms on the compact and orientable