Characteristic Classes and Weyl Tensor: Applications to General Relativity^*

Abstract

In a recent paper, Chern and Simons proved that the Pontrjagin forms of a Riemannian manifold remain invariant under a conformal deformation. We show that these forms can be expressed uniquely in terms of the conformal curvature tensor: this provides a new proof of their result. Similar techniques can be applied to Euler-Poincare characteristic class, as suggested to me by A. Taub. We obtain the following: If the Weyl tensor of a compact space time is of type III of Bel-Petrov, then it cannot carry a perfect fluid + electromagnetic field.