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Fulling, King, Wybourne and Cummings (FKWC) have proposed to expandsystematically the Riemann polynomials encountered in the context of fieldtheories in curved spacetime on standard bases constructed from grouptheoretical considerations. They have also displayed such bases for scalarRiemann polynomials of order eight or less in the derivatives of the metrictensor and for tensorial Riemann polynomials of order six or less. Here weprovide a slightly modified version of the FKWC-bases as well as an importantlist of geometrical relations we have used in recent works. These relations areindependent of the dimension of spacetime. In our opinion, they are helpful toachieve quickly and easily, by hand, very tedious calculations as well as, ofcourse, to provide irreducible expressions for all the results obtained. Theycould be very helpful to people working in gravitational physics and moreparticularly (i) to treat some aspects of the classical theory of gravitationalwaves such as the radiation reaction problem and (ii) to deal withregularization and renormalization of quantum field theories, of stochasticsemiclassical gravity and of higher-order theories of gravity.

FKWC-bases and geometrical identities for classical and quantum field theories in curved spacetime