Two important invariant identities

Two important invariant identities about the products of the Riemann tensor $R_{\alpha \beta \gamma \lambda }$ and the Ricci tensor $R_{\alpha \beta }$ and the scalar curvature R are derived by the technique of Weyl decomposition of the Riemann tensor and by the spinor formalism. These identities are very useful in four dimensions for simplifying the final expression of the $a_3$ coefficient of the scalar fields and for simplifying the evaluation of the vacuum-polarization energy-momentum tensor. This result is of relevance to the work of Jack and Parker on the summed form of the heat kernel.