Geometric structures arising from the deformation of groups of Heisenberg type

TL;DR

This paper introduces a new geometric structure called contact structure of Heisenberg type on manifolds modeled on Heisenberg groups, linking it to the Yamabe problem and providing tools for geometric analysis.

Contribution

It defines a contact structure of Heisenberg type, establishes a natural connection, and formulates the Yamabe problem in this new geometric setting.

Findings

Defined contact structures of Heisenberg type on manifolds

Derived formulas for conformal change of scalar curvature

Formulated the Yamabe problem for these manifolds

Abstract

Motivated by the desire of finding a geometric interpretation to the Yamabe equation on groups of Heisenberg type, we define a geometric structure on manifolds modelled locally on these groups, which we call contact structure of Heisenberg type. In the case of the Heisenberg group is equivalent to contact Riemannian manifolds. We define a natural connection on these structures, we compute the formula for the conformal change of scalar curvature, and introduce the Yamabe problem for these manifods.