The decomposition of global conformal invariants VI: The proof of the proposition on local Riemannian invariants
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.
The Fredholm property of operator, essentially unitary and essentially normal composition are important. Which Fredholm operator are reversible operator is the key problem. The main purpose of this paper is to study Fredholm, essentially unitary and essentially normal composition operators on L~2(u) when the underlying measure space (X,φ,u) is non-atomic. The existence condition of Fredholm operator is discussed.