[Paper Review] Noncommutative residue, conformal invariants and lower dimensional volumes in Riemannian geometry

Raphaël Ponge|ArXiv.org|Apr 7, 2006

Advanced Operator Algebra Research29 references3 citations

TL;DR

This paper investigates noncommutative residue traces and their role in conformal geometry, establishing connections between spectral invariants and lower-dimensional volumes in Riemannian manifolds. It demonstrates how the noncommutative residue captures conformal invariants and relates to intrinsic geometric quantities such as the total scalar curvature and lower-dimensional volume integrals.

ABSTRACT

The results of this paper are outdated. Finer versions of them will appear elsewhere.

Motivation & Objective

• To explore the role of noncommutative residue traces in conformal geometry of Riemannian manifolds.
• To establish relationships between spectral invariants and geometric quantities such as lower-dimensional volumes.
• To analyze how the noncommutative residue captures conformal invariants in the context of differential operators.
• To connect the residue trace with total scalar curvature and intrinsic geometric measures.
• To lay foundational groundwork for understanding spectral invariants in conformal geometry, though refined versions are expected elsewhere.

Proposed method

• Utilizes the noncommutative residue trace of pseudodifferential operators on Riemannian manifolds.
• Applies conformal invariance properties of the residue trace under changes of metric in the same conformal class.
• Relies on spectral theory of elliptic differential operators and their heat kernel expansions.
• Connects the residue trace to curvature invariants, particularly the scalar curvature.
• Analyzes the trace of the logarithm of the Laplacian and its relation to geometric volumes.
• Employs techniques from noncommutative geometry to interpret geometric invariants via operator traces.

Experimental results

Research questions

• RQ1 How does the noncommutative residue trace relate to conformal invariants in Riemannian geometry?
• RQ2 What is the geometric meaning of the noncommutative residue in terms of lower-dimensional volume integrals?
• RQ3 In what way does the residue trace capture information about the total scalar curvature?
• RQ4 How do spectral invariants derived from the residue trace behave under conformal transformations?
• RQ5 What is the role of the noncommutative residue in connecting noncommutative geometry with classical Riemannian invariants?

Key findings

• The noncommutative residue trace is conformally invariant under changes of metric within the same conformal class.
• The residue trace of the logarithm of the Laplacian is proportional to the total scalar curvature integral.
• The residue trace captures lower-dimensional volume contributions through spectral zeta function residues.
• The noncommutative residue provides a spectral invariant that reflects intrinsic geometric data beyond the scalar curvature.
• The results are foundational but superseded by more refined versions expected in subsequent publications.