Maximal Tori in Contactomorphism Groups

TL;DR

This paper introduces a scheme linking conjugacy classes of tori in contactomorphism groups to transverse almost complex structures, and constructs Sasaki cones for contact structures of K-contact type, revealing a bouquet configuration.

Contribution

It establishes a novel connection between tori conjugacy classes and transverse structures, and constructs Sasaki cones forming a bouquet for K-contact contact structures.

Findings

Conjugacy classes of tori correspond to transverse almost complex structures.

Sasaki cones are associated with Reeb type tori containing Reeb vector fields.

A bouquet of Sasaki cones is formed for K-contact structures.

Abstract

I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-contact type one obtains a configuration of Sasaki cones called a bouquet such that each Sasaki cone is associated with a conjugacy class of tori of Reeb type.