Five dimensional K-contact manifolds of rank 2

TL;DR

This paper classifies 5-dimensional K-contact manifolds of rank 2 using Morse theory, establishing their relation to graphs of isotropy data, and explores conditions for their Sasakian structure and toric properties.

Contribution

It provides a classification framework for 5D K-contact manifolds of rank 2, including their isomorphism classes, contact blow-up/down relations, and conditions for toric structures.

Findings

Classification via isotropy data graphs

Existence of Sasakian metrics on all such manifolds

Conditions for being contact toric

Abstract

A K-contact manifold is a smooth manifold M with a contact form whose Reeb flow preserves a Riemannian metric on M. Main examples are Sasakian manifolds. Our results in this paper are four results i), ii), iii) and iv) below obtained by the application of the Morse theory to the contact moment maps on closed 5-dimensional K-contact manifolds of rank 2. 5-dimensional K-contact manifolds of rank 2 have the lowest dimension among K-contact manifolds with nontrivial contact moment maps which are not toric. i) Correspondence between the isomorphism classes of K-contact manifolds and graphs of isotropy data, ii) Classification of K-contact manifolds up to contact blowing up and down, iii) Every closed 5-dimensional K-contact manifold of rank 2 has a Sasakian metric, iv) A sufficient condition for closed 5-dimensional K-contact manifolds of rank 2 to be contact toric.