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

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

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

${\displaystyle \eta }$

${\displaystyle \varphi }$

${\displaystyle M,}$

${\displaystyle v}$

${\displaystyle T_{p}M=Q_{p}\oplus \left\{k\xi _{p}:k\in \mathbb {R} \right\},}$

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

${\displaystyle v}$

${\displaystyle T_{p}M.}$

${\displaystyle (\xi ,\eta ,\varphi )}$

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

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