Canonical connection on contact manifolds

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 $(Q,\lambda,J)$.