Contact triad connection of contact manifolds in almost contact moving frame

TL;DR

This paper simplifies and canonically constructs the contact triad connection on contact manifolds, facilitating analysis of contact instanton equations and related geometric PDEs.

Contribution

It provides a more straightforward, canonical construction of the contact triad connection using the almost contact moving frame, enhancing geometric analysis tools.

Findings

Simplified and canonical construction of the contact triad connection.

Enhanced tools for analyzing contact instanton equations.

Applications to pseudoholomorphic curves and asymptotic operators.

Abstract

The notion of \emph{contact triad connection} on contact triads $(Q,\lambda,J_\xi)$ was introduced by Wang and the present author in early 2010's from scratch as the contact analog to the canonical connection of an almost K\"ahler manifolds. The connection facilitates the study of analysis of the contact instanton equation which ranges from local elliptic priori estimates, for both the interior [OW2,OW3] and the boundary [OY], to the generic perturbation theory of the asymptotic operators [KO], which also encompasses the case of pseudoholomorphic curves on noncompact symplectic manifolds with cylindrical ends [OK]. The main purpose of the present paper is to give a simpler and more canonical construction of the contact triad connection by first giving its characterization in terms of the associated \emph{almost contact structure} and then providing its construction using the almost contact moving frame in the framework of contact geometry.