Almost-contact manifold

Last updated June 28, 2023

In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a smooth manifold. Such structures were introduced by Shigeo Sasaki in 1960.

Precisely, given a smooth manifold $M,$ an almost-contact structure consists of a hyperplane distribution $Q,$ an almost-complex structure $J$ on $Q,$ and a vector field $\xi$ which is transverse to $Q.$ That is, for each point $p$ of $M,$ one selects a codimension-one linear subspace $Q_{p}$ of the tangent space $T_{p}M,$ a linear map $J_{p}:Q_{p}\to Q_{p}$ such that $J_{p}\circ J_{p}=-\operatorname {id} _{Q_{p}},$ and an element $\xi _{p}$ of $T_{p}M$ which is not contained in $Q_{p}.$

Given such data, one can define, for each $p$ in $M,$ a linear map $\eta _{p}:T_{p}M\to \mathbb {R}$ and a linear map $\varphi _{p}:T_{p}M\to T_{p}M$ by

{\begin{aligned}\eta _{p}(u)&=0{\text{ if }}u\in Q_{p}\\\eta _{p}(\xi _{p})&=1\\\varphi _{p}(u)&=J_{p}(u){\text{ if }}u\in Q_{p}\\\varphi _{p}(\xi )&=0.\end{aligned}}

This defines a one-form $\eta$ and (1,1)-tensor field $\varphi$ on $M,$ and one can check directly, by decomposing $v$ relative to the direct sum decomposition $T_{p}M=Q_{p}\oplus \left\{k\xi _{p}:k\in \mathbb {R} \right\},$ that

{\begin{aligned}\eta _{p}(v)\xi _{p}&=\varphi _{p}\circ \varphi _{p}(v)+v\end{aligned}}

for any $v$ in $T_{p}M.$ Conversely, one may define an almost-contact structure as a triple $(\xi ,\eta ,\varphi )$ which satisfies the two conditions

$\eta _{p}(v)\xi _{p}=\varphi _{p}\circ \varphi _{p}(v)+v$ for any $v\in T_{p}M$
$\eta _{p}(\xi _{p})=1$

Then one can define $Q_{p}$ to be the kernel of the linear map $\eta _{p},$ and one can check that the restriction of $\varphi _{p}$ to $Q_{p}$ is valued in $Q_{p},$ thereby defining $J_{p}.$

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References

David E. Blair. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhäuser Boston, Ltd., Boston, MA, 2010. xvi+343 pp. ISBN 978-0-8176-4958-6, doi : 10.1007/978-0-8176-4959-3
Sasaki, Shigeo (1960). "On differentiable manifolds with certain structures which are closely related to almost contact structure, I". Tohoku Mathematical Journal. 12 (3): 459–476. doi:10.2748/tmj/1178244407.

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