Canonical connection on contact manifolds

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 ${\varphi }^{\ast }\mathrm{\nabla }$ of a triad connection $\mathrm{\nabla }$ becomes the triad connection of the pull-back triad $(Q, \phi^*\lambda, \phi^J)$foranydiffeomorphism$\phi:Q \to Q$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 ${\mathrm{\nabla }}^{\pi }$ on the Hermitian vector bundle $(\xi ,J,g_{\xi })$ with $g_{\xi }=d\lambda (\cdot ,J\cdot ){\mathrm{\mid }}_{\xi }$, 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 ${\bar{\mathrm{\partial }}}^{\pi }w=0,d(w^{\ast }\lambda \circ j)=0$ in the contact triad $(Q,\lambda ,J)$.