Bianchi identities for the Riemann and Weyl tensors

TL;DR

This paper revisits Bianchi identities for Riemann and Weyl tensors using advanced algebraic and differential methods, revealing their order and structure across different dimensions, with implications for geometric analysis.

Contribution

It provides a unified algebraic and differential framework to explicitly describe Bianchi identities for Riemann and Weyl tensors in arbitrary dimensions, correcting previous inconsistencies.

Findings

Bianchi identities for Riemann tensor are first order and dimension-independent.

Weyl tensor Bianchi identities are first order for n ≥ 5, second order in n=4.

Identifies discrepancies in classical treatments of Bianchi identities, especially in dimension 4.

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.