Analytic torsion of generic rank two distributions in dimension five

TL;DR

This paper introduces an analytic torsion for the Rumin complex on 5-manifolds with rank two distributions, exploring its properties, anomaly formulas, and relation to Ray-Singer torsion on specific nilmanifolds.

Contribution

It defines a new analytic torsion for a geometric structure on 5-manifolds and establishes its fundamental properties and connections to existing torsion invariants.

Findings

Torsion behaves consistently with Poincare duality and coverings

Derived anomaly formulas relating torsion to local geometric data

Showed equivalence with Ray-Singer torsion on certain nilmanifolds

Abstract

We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincare duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds we are able to show that this torsion coincides with the Ray-Singer analytic torsion, up to a constant.