Canonical connection on contact manifolds

Abstract

We introduce a canonical affine connection on the contact manifold $(Q,ξ)$, which is associated to each contact triad $(Q,λ,J)$ where $λ$ is a contact form and $J:ξ\to ξ$ is an endomorphism with $J^2 = -id$ compatible to $dλ$. We call it the \emph{contact triad connection} of $(Q,λ,J)$ and prove its existence and uniqueness. The connection is canonical in that the pull-back connection $φ^*\nabla$ of a triad connection $\nabla$ becomes the triad connection of the pull-back triad $(Q, φ^*λ, φ^*J)$ for any diffeomorphism $φ:Q \to Q$ satisfying $φ^*λ= λ$ (sometimes called a strict contact diffeomorphism). It also preserves both the triad metric $$ g_{(λ,J)} = dλ(\cdot, J\cdot) + λ\otimes λ$$ and $J$ regarded as an endomorphism on $TQ = \mathbb R\{X_λ\}\oplus ξ$, and is characterized by its torsion properties and the requirement that the contact form $λ$ be holomorphic in the $CR$-sense. In particular, the connection restricts to a Hermitian connection $\nabla^π$ on the Hermitian vector bundle $(ξ,J,g_ξ)$ with $g_ξ= dλ(\cdot, J\cdot)|_ξ$, which we call the \emph{contact Hermitian connection} of $(ξ,J,g_ξ)$. 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 $\overline{\partial}^πw = 0, \, d(w^*λ\circ j) = 0$ in the contact triad $(Q,λ,J)$.