The decomposition of global conformal invariants IV: A proposition on local Riemannian invariants

TL;DR

This paper proves algebraic properties of local Riemannian invariants crucial for understanding the structure of global conformal invariants, confirming a conjecture about their decomposition into fundamental geometric components.

Contribution

It establishes key algebraic results on local Riemannian invariants that support the decomposition of global conformal invariants, advancing the proof of Deser and Schwimmer's conjecture.

Findings

Proved algebraic properties of local Riemannian invariants.

Confirmed the structure of global conformal invariants as a combination of fundamental terms.

Extended the theoretical framework for conformal geometry and invariants.

Abstract

This is the fourth 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 lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these were used extensively in [3,4]. These results may be of independent interest, applicable to related problems.