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 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
This defines a one-form
and (1,1)-tensor field
on
and one can check directly, by decomposing
relative to the direct sum decomposition
that
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. 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.
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