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).