Analytic Torsion of Generic Rank Two Distributions in Dimension Five

Area of Science:

• Differential Geometry
• Geometric Analysis
• Topology

Background:

• Analytic torsion is a key invariant in spectral geometry.
• Rumin complexes are essential for studying differential forms on manifolds with non-smooth structures.
• Rank two distributions on 5-manifolds present complex geometric challenges.

Purpose of the Study:

• To define and analyze a novel analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds.
• To investigate the behavior of this torsion under Poincaré duality and finite coverings.
• To establish anomaly formulas relating the torsion to local geometric quantities.

Main Methods:

• Construction of an analytic torsion for the Rumin complex.
• Verification of invariance properties under Poincaré duality and finite coverings.
• Derivation of anomaly formulas using integration of local quantities.

Main Results:

• The proposed analytic torsion is well-defined and exhibits expected behavior under standard geometric transformations.
• Anomaly formulas were established, connecting the torsion to the sub-Riemannian metric and the 2-plane bundle.
• For specific nilmanifolds, the analytic torsion was shown to be equivalent to the Ray-Singer analytic torsion.

Conclusions:

• The developed analytic torsion provides a valuable tool for studying the geometry of rank two distributions.
• The anomaly formulas offer new perspectives on the relationship between global and local invariants in sub-Riemannian geometry.
• The coincidence with Ray-Singer torsion for nilmanifolds validates the proposed invariant and suggests broader connections.