Canonical connection on contact manifolds

TL;DR

This paper introduces a unique, canonical affine connection on contact manifolds compatible with the contact structure, which simplifies calculations and is fundamental for studying contact instantons.

Contribution

The authors define and prove the existence and uniqueness of a canonical contact triad connection that preserves the contact structure and metric, and characterizes it via torsion and holomorphic conditions.

Findings

Defines the contact triad connection and proves its uniqueness.

Shows the connection preserves the triad metric and J.

Simplifies tensorial calculations in contact instanton theory.

Abstract

We introduce a canonical affine connection on the contact manifold $(Q,\xi)$, which is associated to each contact triad $(Q,\lambda,J)$ where $\lambda$ is a contact form and $J:\xi \to \xi$ is an endomorphism with $J^2 = -id$ compatible to $d\lambda$. We call it the \emph{contact triad connection} of $(Q,\lambda,J)$ and prove its existence and uniqueness. The connection is canonical in that the pull-back connection $\phi^*\nabla$ of a triad connection $\nabla$ becomes the triad connection of the pull-back triad $(Q, \phi^*\lambda, \phi^*J)$ for any diffeomorphism $\phi:Q \to Q$ satisfying $\phi^*\lambda = \lambda$ (sometimes called a strict contact diffeomorphism). It also preserves both the triad metric $$ g_{(\lambda,J)} = d\lambda(\cdot, J\cdot) + \lambda \otimes \lambda $$ and $J$ regarded as an endomorphism on $TQ = \mathbb R\{X_\lambda\}\oplus \xi$, and is characterized by its torsion properties and the requirement that the contact form $\lambda$ be holomorphic in the $CR$-sense. In particular, the connection restricts to a Hermitian connection $\nabla^\pi$ on the Hermitian vector bundle $(\xi,J,g_\xi)$ with $g_\xi = d\lambda(\cdot, J\cdot)|_{\xi}$, which we call the \emph{contact Hermitian connection} of $(\xi,J,g_\xi)$. These connections greatly simplify tensorial calculations in the sequels \cite{oh-wang1}, \cite{oh-wang2} performed in the authors' analytic study of the map $w$, called contact instantons, which satisfy the nonlinear elliptic system of equations $\overline{\partial}^\pi w = 0, \, d(w^*\lambda \circ j) = 0$ in the contact triad $(Q,\lambda,J)$.