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
, 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
TpM=Qp ⊕ \left\{k\xip:k\in\R\right\}
, that
for any
in
. Conversely, one may define an almost-contact structure as a triple
which satisfies the two conditions
ηp(v)\xip=\varphip\circ\varphip(v)+v
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.,
- 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 .
