Harmonic forms and the Rumin complex on Sasakian manifolds

TL;DR

This paper establishes a relationship between harmonic forms and the Rumin complex on compact Sasakian manifolds, providing new insights into their structure and analytic torsion.

Contribution

It proves the equivalence of harmonic forms for the Rumin and Hodge-de Rham Laplacians on Sasakian manifolds and relates the analytic torsion to the Reeb vector field.

Findings

Kernel of Rumin Laplacian matches Hodge-de Rham Laplacian on Sasakian manifolds

Harmonic forms are primitive and coincide with the sub-Riemann limit

Analytic torsion function expressed via Reeb vector field

Abstract

We show that the kernel of the Rumin Laplacian agrees with that of the Hodge-de Rham Laplacian on compact Sasakian manifolds. As a corollary, we obtain another proof of primitiveness of harmonic forms. Moreover, the space of harmonic forms coincides with the sub-Riemann limit of Hodge-de Rham Laplacian when its limit converges. Finally, we express the analytic torsion function associated with the Rumin complex in terms of the Reeb vector field.