Symmetry Via Lie Algebra Cohomology

Abstract

The Killing operator on a Riemmanian manifold is a linear differetial operator on vector fields whose kernel provides the infinitesimal Riemannian symmetries. The Killing operator is best understood in terms of its prolongation, which entails some simple tensor identities. These simple identities can be viewed as arising from the vanishing of certain Lie algebra cohomologies. The point is that this case provides a model for other more complicated operators similiarly concerned with symmetry.