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 $J_p \circ J_p = - \operatorname_,$ 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
$\begin \eta_p(u)&=0\textu\in Q_p\\ \eta_p(\xi_p)&=1\\ \varphi_p(u)&=J_p(u)\textu\in Q_p\\ \varphi_p(\xi)&=0. \end$
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 $T_p M = Q_p \oplus \left\,$ that
$\begin \eta_p(v) \xi_p &= \varphi_p \circ \varphi_p(v) + v \end$
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 $J_p.$

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. ,
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{{Manifolds

Differential geometry
Smooth manifolds