Canonical Connection on Contact Manifolds

  • Yong-Geun Oh
  • Rui Wang
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)


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).


Vector Bundle Contact Form Contact Manifold Torsion Tensor Hermitian Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Luigi Vezzoni and Liviu Nocolaescu for alerting their works [11, 19] on special connections on contact manifolds after the original version of the present paper was posted in the arXiv e-print. We also thank them for helpful discussions. The present work was supported by the IBS project IBS-R003-D1.


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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Center for Geometry and PhysicsInstitute for Basic Science (IBS)Pohang-siKorea
  2. 2.POSTECHPohangKorea

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