Almost-contact manifold

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 $P$ , an almost-complex structure $J$ on $Q$ , and a vector field $ξ$ 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 $φ p : Q p \to Q p$ such that $J p \circ J p = -id Q p$ , and an element $ξ p$ 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 $η p : T p M \to ℝ$ and a linear map $φ p : T p M \to T p M$ 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 $η$ and (1,1)-tensor field $φ$ on $M$ , and one can check directly, by decomposing $v$ relative to the direct sum decomposition $T p M = Q p \oplus {k ξ p : k \in ℝ$ }, 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 $(ξ, η, φ)$ 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 $η p$ , and one can check that the restriction of $φ p$ to $Q p$ is valued in $Q p$ , thereby defining $J p$ .

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
Shigeo Sasaki. On differentiable manifolds with certain structures which are closely related to almost contact structure. I. Tohoku Math. J. (2) 12 (1960), 459–476.

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