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_pM, a linear map φ_p : Q_p → Q_p such that J_p ∘ J_p = −id_Qp, and an element ξ_p of T_pM which is not contained in Q_p.

Given such data, one can define, for each p in M, a linear map η_p : T_pM → ℝ and a linear map φ_p : T_pM → T_pM by

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

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_pM = Q_p ⊕ {kξ_p : k ∈ ℝ}, that

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

for any v in T_pM. Conversely, one may define an almost-contact structure as a triple (ξ, η, φ) which satisfies the two conditions

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

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.