Benjamin McMillan, The Range of the Killing Operator

The Killing operator in (semi) Riemannian geometry has well understood kernel: the infinitesimal symmetries of a given metric. At the next level, the range of the Killing operator can be interpreted as those perturbations of the metric that result from a mere change of coordinates—in contexts like general relativity, the trivial perturbations. The range of the Killing operator, albeit important, has not been as well understood as its kernel. I will describe how one can use the projective nature of the Killing operator to reduce the question to that of the range of a connection on an associated bundle, and then how one can seek to understand the range of an arbitrary connection. This process yields a fairly complete answer on locally symmetric spaces.

Based on joint work with Federico Costanza, Mike Eastwood, and Thomas Leistner.