2012/12/19 by Yong‐Geun Oh, Rui Wang, Oh, Yong-Geun +1
Mathematics · #53D42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.1212.4817
openalex publication_date 2012/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→ ξ is an endomorphism with J2 = -id compatible to dλ. We call it 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, ϕ^*λ, ϕ^*J) for any diffeomorphism ϕ:Q → Q satisfying ϕ^*λ= λ (sometimes called a strict contact diffeomorphism). It also preserves both the triad metric g(λ,J) = dλ(⋅, J⋅) + λ⊗ λ and J regarded as an endomorphism on TQ = \mathbb R\Xλ\⊕ 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 ∇π on the Hermitian vector bundle (ξ,J,gξ) with gξ= dλ(⋅, J⋅)|ξ, which we call the contact Hermitian connection of (ξ,J,gξ). These connections greatly simplify tensorial calculations in the sequels \citeoh-wang1, \citeoh-wang2 performed in the authors' analytic study of the map w, called contact instantons, which satisfy the nonlinear elliptic system of equations ∂πw = 0, d(w^*λ∘ j) = 0 in the contact triad (Q,λ,J).