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
, and a vector field
which is transverse to
, one selects a codimension-one linear subspace
of the tangent space
, a linear map
id
{\displaystyle J_{p}\circ J_{p}=-\operatorname {id} _{Q_{p}}}
Given such data, one can define, for each
, a linear map
and a linear map
φ
φ
φ
This defines a one-form
and (1,1)-tensor field
φ
, and one can check directly, by decomposing
relative to the direct sum decomposition
φ
Conversely, one may define an almost-contact structure as a triple
( ξ , η , φ )
which satisfies the two conditions Then one can define
to be the kernel of the linear map
, and one can check that the restriction of
, thereby defining