Pontrjagin numbers and pseudo-riemannian geometry

Abstract

An integral formula for the Pontrjagin numbers of a compact orientable real 4k dimensional differentiable manifold which has a pseudo-Riemannian metric is derived. This formula allows the Pontrjagin numbers to be expressed in terms of the index, or signature, of the differentiable manifold. The application of these formulae to the four dimensional Lorentzian manifolds of the general theory of relativity is discussed. A corresponding formula for the Chern numbers of a complex differentiable manifold with a Hermitian metric is also given.