Bianchi identities for the Riemann and Weyl tensors

Abstract

The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems, formal integrability) and Algebraic Analysis (homological algebra, differential modules, duality). In particular, we prove that the n 2 (n 2 -- 1)(n -- 2)/24 generating Bianchi identities for the Riemann tensor are first order and can be easily described by means of the Spencer cohomology of the first order Killing symbol in arbitrary dimension n $\ge$ 2. Similarly, the n(n 2 -- 1)(n + 2)(n -- 4)/24 generating Bianchi identities for the Weyl tensor are first order and can be easily described by means of the Spencer cohomology of the first order conformal Killing symbol in arbitrary dimension n $\ge$ 5. As A MOST SURPRISING RESULT, the 9 generating Bianchi identities for the Weyl tensor are of second order in dimension n = 4 while the analogue of the Weyl tensor has 5 components of third order in the metric with 3 first order generating Bianchi identities in dimension n = 3. The above results, which could not be obtained otherwise, are valid for any non-degenerate metric of constant riemannian curvature and do not depend on any conformal factor. They are checked in an Appendix produced by Alban Quadrat (INRIA, Lille) by means of computer algebra. We finally explain why the work of Lanczos and followers is not coherent with these results and must therefore be also revisited.