Almost-contact manifold

Short description: Geometric structure on a smooth 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 [math]\displaystyle{ M, }[/math] an almost-contact structure consists of a hyperplane distribution [math]\displaystyle{ Q, }[/math] an almost-complex structure [math]\displaystyle{ J }[/math] on [math]\displaystyle{ Q, }[/math]and a vector field [math]\displaystyle{ \xi }[/math] which is transverse to [math]\displaystyle{ Q. }[/math] That is, for each point [math]\displaystyle{ p }[/math] of [math]\displaystyle{ M, }[/math] one selects a codimension-one linear subspace [math]\displaystyle{ Q_p }[/math] of the tangent space [math]\displaystyle{ T_p M, }[/math] a linear map [math]\displaystyle{ J_p : Q_p \to Q_p }[/math] such that [math]\displaystyle{ J_p \circ J_p = - \operatorname{id}_{Q_p}, }[/math] and an element [math]\displaystyle{ \xi_p }[/math] of [math]\displaystyle{ T_p M }[/math] which is not contained in [math]\displaystyle{ Q_p. }[/math]

Given such data, one can define, for each [math]\displaystyle{ p }[/math] in [math]\displaystyle{ M, }[/math] a linear map [math]\displaystyle{ \eta_p : T_p M \to \R }[/math] and a linear map [math]\displaystyle{ \varphi_p : T_p M \to T_p M }[/math] by [math]\displaystyle{ \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} }[/math] This defines a one-form [math]\displaystyle{ \eta }[/math] and (1,1)-tensor field [math]\displaystyle{ \varphi }[/math] on [math]\displaystyle{ M, }[/math] and one can check directly, by decomposing [math]\displaystyle{ v }[/math] relative to the direct sum decomposition [math]\displaystyle{ T_p M = Q_p \oplus \left\{ k \xi_p : k \in \R \right\}, }[/math] that [math]\displaystyle{ \begin{align} \eta_p(v) \xi_p &= \varphi_p \circ \varphi_p(v) + v \end{align} }[/math] for any [math]\displaystyle{ v }[/math] in [math]\displaystyle{ T_p M. }[/math] Conversely, one may define an almost-contact structure as a triple [math]\displaystyle{ (\xi, \eta, \varphi) }[/math] which satisfies the two conditions

• [math]\displaystyle{ \eta_p(v) \xi_p = \varphi_p \circ \varphi_p(v) + v }[/math] for any [math]\displaystyle{ v \in T_p M }[/math]
• [math]\displaystyle{ \eta_p(\xi_p) = 1 }[/math]

Then one can define [math]\displaystyle{ Q_p }[/math] to be the kernel of the linear map [math]\displaystyle{ \eta_p, }[/math] and one can check that the restriction of [math]\displaystyle{ \varphi_p }[/math] to [math]\displaystyle{ Q_p }[/math] is valued in [math]\displaystyle{ Q_p, }[/math] thereby defining [math]\displaystyle{ J_p. }[/math]

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
• Sasaki, Shigeo (1960). "On differentiable manifolds with certain structures which are closely related to almost contact structure, I". Tohoku Mathematical Journal 12 (3): 459–476. doi:10.2748/tmj/1178244407.