Английская Википедия:Almost-contact manifold
Шаблон:Short description Шаблон:No footnotes 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>M,</math> an almost-contact structure consists of a hyperplane distribution <math>Q,</math> an almost-complex structure <math>J</math> on <math>Q,</math>and a vector field <math>\xi</math> which is transverse to <math>Q.</math> That is, for each point <math>p</math> of <math>M,</math> one selects a codimension-one linear subspace <math>Q_p</math> of the tangent space <math>T_p M,</math> a linear map <math>J_p : Q_p \to Q_p</math> such that <math>J_p \circ J_p = - \operatorname{id}_{Q_p},</math> and an element <math>\xi_p</math> of <math>T_p M</math> which is not contained in <math>Q_p.</math>
Given such data, one can define, for each <math>p</math> in <math>M,</math> a linear map <math>\eta_p : T_p M \to \R</math> and a linear map <math>\varphi_p : T_p M \to T_p M</math> by <math display=block>\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>\eta</math> and (1,1)-tensor field <math>\varphi</math> on <math>M,</math> and one can check directly, by decomposing <math>v</math> relative to the direct sum decomposition <math>T_p M = Q_p \oplus \left\{ k \xi_p : k \in \R \right\},</math> that <math display=block>\begin{align} \eta_p(v) \xi_p &= \varphi_p \circ \varphi_p(v) + v \end{align}</math> for any <math>v</math> in <math>T_p M.</math> Conversely, one may define an almost-contact structure as a triple <math>(\xi, \eta, \varphi)</math> which satisfies the two conditions
- <math>\eta_p(v) \xi_p = \varphi_p \circ \varphi_p(v) + v</math> for any <math>v \in T_p M</math>
- <math>\eta_p(\xi_p) = 1</math>
Then one can define <math>Q_p</math> to be the kernel of the linear map <math>\eta_p,</math> and one can check that the restriction of <math>\varphi_p</math> to <math>Q_p</math> is valued in <math>Q_p,</math> thereby defining <math>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. Шаблон:ISBN, Шаблон:Doi Шаблон:Closed access
- Шаблон:Cite journal
