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.
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
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.