On differential characteristic classes of metrics and connections

A differential characteristic class of a geometric quantity (e.g., Riemannian or Kähler metric, connection, etc.) on a smooth manifold is a closed differential form whose components are expressed in the components of the given geometric quantity and in their partial derivatives in local coordinates via algebraic formulas independent of the choice of coordinates, and whose cohomology class is stable under deformations of the given quantity. In this note, we present a short proof of the theorem of P. Gilkey on characteristic classes of Riemannian metrics, which is based on the method of invariant-theoretic reduction developed by P. I. Katsylo and D. A. Timashev, and generalize this result to Kähler metrics and connections.