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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 an almost-contact structure consists of a hyperplane distribution an almost-complex structure on and a vector field which is transverse to That is, for each point of one selects a codimension-one linear subspace of the tangent space a linear map such that and an element of which is not contained in
Given such data, one can define, for each in a linear map and a linear map by
for any in Conversely, one may define an almost-contact structure as a triple which satisfies the two conditions
for any
Then one can define to be the kernel of the linear map and one can check that the restriction of to is valued in thereby defining
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. ISBN978-0-8176-4958-6, doi:10.1007/978-0-8176-4959-3
Shigeo Sasaki. On differentiable manifolds with certain structures which are closely related to almost contact structure. I. Tohoku Math. J. (2) 12 (1960), 459–476.