Canonical Connection on Contact Manifolds

Abstract

We introduce a family of canonical affine connections 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 → ξ is an endomorphism with \(J^{2} = -id\) compatible to d λ. We call a particular one in this family the contact triad connection of (Q, λ, J) and prove its existence and uniqueness. The connection is canonical in that the pull-back connection ϕ ∗∇ of a triad connection ∇ becomes the triad connection of the pull-back triad \((Q,\phi ^{{\ast}}\lambda,\phi ^{{\ast}}J)\) for any diffeomorphism ϕ: Q → Q. It also preserves both the triad metric \(g:= d\lambda (\cdot,J\cdot ) +\lambda \otimes \lambda\) 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 λ be holomorphic in the CR-sense. In particular, the connection restricts to a Hermitian connection ∇ π on the Hermitian vector bundle (ξ, J, g ξ ) with \(g_{\xi } = d\lambda (\cdot,J\cdot )\vert _{\xi }\), which we call the contact Hermitian connection of (ξ, J, g ξ ). These connections greatly simplify tensorial calculations in the sequels (Oh and Wang, The Analysis of Contact Cauchy-Riemann maps I: a priori Ck estimates and asymptotic convergence, preprint. arXiv:1212.5186, 2012; Oh and Wang, Analysis of contact instantons II: exponential convergence for the Morse-Bott case, preprint. arXiv:1311.6196, 2013) performed in the authors’ analytic study of the map w, called contact instantons, which satisfy the nonlinear elliptic system of equations $$\displaystyle{\overline{\partial }^{\pi }w = 0,\,d(w^{{\ast}}\lambda \circ j) = 0}$$ of the contact triad (Q, λ, J).