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 $$Q,$$ 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 $$\eta_p : T_p M \to \R$$ and a linear map $$\varphi_p : T_p M \to T_p M$$ by $$\begin{align} \eta_p(u)&=0\text{ if }u\in Q_p\\ \eta_p(\xi_p)&=1\\ \varphi_p(u)&=J_p(u)\text{ if }u\in Q_p\\ \varphi_p(\xi)&=0. \end{align}$$ 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 $$T_p M = Q_p \oplus \left\{ k \xi_p : k \in \R \right\},$$ that $$\begin{align} \eta_p(v) \xi_p &= \varphi_p \circ \varphi_p(v) + v \end{align}$$ 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 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 $$J_p.$$
 * $$\eta_p(v) \xi_p = \varphi_p \circ \varphi_p(v) + v$$ for any $$v \in T_p M$$
 * $$\eta_p(\xi_p) = 1$$