Almost-contact manifold explained

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

M, an almost-contact structure consists of a hyperplane distribution Q, an almost-complex structure J on Q, and a vector field \xi which is transverse to Q. That is, for each point p of M, one selects a codimension-one linear subspace Q _{p} of the tangent space T _{p} M, a linear map J _{p} : Q _{p} \to Q _{p} such that J _{p} \circ J _{p} = - \operatorname{id} _{Q _{p}} , and an element \xi _{p} of T _{p} M which is not contained in Q _{p.}

Given such data, one can define, for each

p in M, a linear map η _{p} : T _{p} M \to \R and a linear map \varphi _{p} : T _{p} M \to T _{p} M byThis defines a one-form η and (1,1)-tensor field \varphi on M, and one can check directly, by decomposing v relative to the direct sum decomposition T _{p} M = Q _{p} ⊕ \left\{ k \xi _{p} : k \in \R \right\}, thatfor any v in T _{p} M. Conversely, one may define an almost-contact structure as a triple (\xi, η , \varphi) which satisfies the two conditionsη _{p(v)} \xi _{p} = \varphi _{p} \circ \varphi _{p(v)} + v for any v \in T _{p} M η _{p(\xi} _{p)} = 1 Then one can define Q _{p} to be the kernel of the linear map η _{p,} and one can check that the restriction of \varphi _{p} to Q _{p} is valued in Q _{p,} thereby defining J _{p.}

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.,
• 10.2748/tmj/1178244407. On differentiable manifolds with certain structures which are closely related to almost contact structure, I . 1960 . Sasaki . Shigeo . Tohoku Mathematical Journal . 12 . 3. 459–476. free .