Sasakian metrics with an additional contact structure

TL;DR

This paper investigates conditions under which Sasakian metrics can support an additional compatible contact structure, extending known results primarily in dimensions 3 and 5, and characterizing the manifold's geometry.

Contribution

It extends previous results on Sasakian metrics with additional contact structures, especially in low dimensions, providing new classifications and conditions.

Findings

Manifolds are 3-Sasakian or spheres with constant curvature under certain conditions

Extensions of known results are achieved in dimensions 3 and 5

Additional compatible contact structures impose strong geometric constraints

Abstract

The question of whether a Sasakian metric can admit an additional compatible (K-)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold must be 3-Sasakian or an odd dimensional sphere with constant curvature. Some extensions of this result are obtained, mainly in dimensions 3 and 5.