Adapted connections on metric contact manifolds

TL;DR

This paper characterizes the space of adapted connections on metric contact manifolds by analyzing their torsion tensors, providing new insights into their structure and related geometric operators.

Contribution

It offers a detailed description of adapted connections via torsion tensor decomposition and characterizes the generalized Tanaka-Webster connection and Dirac operators.

Findings

Complete description of adapted connections through torsion tensor splitting

Characterization of the generalized Tanaka-Webster connection

Description of Dirac operators of adapted connections

Abstract

In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms, which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to completely describe the space of adapted connections. We use this description to obtain characterizations of the generalized Tanaka-Webster connection and to describe the Dirac operators of adapted connections.