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- W4300889554 abstract "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 $dlambda$. 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 $phi^*nabla$ of a triad connection $nabla$ becomes the triad connection of the pull-back triad $(Q, phi^*lambda, phi^*J)$ for any diffeomorphism $phi:Q to Q$ satisfying $phi^*lambda = lambda$ (sometimes called a strict contact diffeomorphism). It also preserves both the triad metric $$ g_{(lambda,J)} = dlambda(cdot, Jcdot) + lambda otimes lambda $$ 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 $nabla^pi$ on the Hermitian vector bundle $(xi,J,g_xi)$ with $g_xi = dlambda(cdot, Jcdot)|_{xi}$, 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 $overline{partial}^pi w = 0, , d(w^*lambda circ j) = 0$ in the contact triad $(Q,lambda,J)$." @default.
- W4300889554 created "2022-10-04" @default.
- W4300889554 creator A5024995818 @default.
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- W4300889554 date "2012-12-19" @default.
- W4300889554 modified "2023-09-27" @default.
- W4300889554 title "Canonical connection on contact manifolds" @default.
- W4300889554 doi "https://doi.org/10.48550/arxiv.1212.4817" @default.
- W4300889554 hasPublicationYear "2012" @default.
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