The decomposition of global conformal invariants VI: The proof of the proposition on local Riemannian invariants

TL;DR

This paper proves a key algebraic proposition about local Riemannian invariants, advancing the resolution of the Deser-Schwimmer conjecture on the structure of global conformal invariants.

Contribution

It provides a crucial algebraic proof that supports the conjecture, with potential applications to related geometric problems.

Findings

Proof of the algebraic proposition on local Riemannian invariants

Supports the conjecture that global conformal invariants decompose into specific geometric components

Establishes a foundation for further analysis of conformal invariants in differential geometry

Abstract

This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper, jointly with [6,7] gives a proof of an algebraic Proposition regarding local Riemannian invariants, which lies at the heart of our resolution of the Deser-Schwimmer conjecture. This algebraic Propositon may be of independent interest, applicable to related problems.