On integral relations for invariants constructed from three Riemann tensors and their applications in quantum gravity

The lowest order quantum corrections to pure gravitation are finite because there exists an integral relation between products of two Riemann tensors (the Gauss–Bonnet theorem). In this article several algebraic and integral relations are determined between products of three Riemann tensors in four- and six-dimensional spacetime. In both cases, one is left with only one invariant when Rμν=0, viz., ℱ (−g)1/2(RαβμνRμνρσRρσ αβ).It is explicitly shown that this invariant does not vanish, even when Rμν=0. Consequently, the two-loop quantum corrections to pure gravitation will only be finite if, due to miraculous cancellation, the coefficient of this invariant vanishes.