In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

. Such structures were introduced by

Shigeo Sasaki Shigeo Sasaki () (18 November 1912 Yamagata Prefecture, Japan – 14 August 1987 Tokyo) was a Japanese mathematician working on differential geometry who introduced Sasaki manifolds. He retired from Tohoku University , or is a Japanese nat ...

in 1960. Precisely, given a smooth manifold

M,

an almost-contact structure consists of a hyperplane distribution

Q,

almost-complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not compl ...

J

Q,

and a vector field

\xi

which is transverse to

Q.

That is, for each point

p

M,

one selects a codimension-one linear subspace

Q_p

of the

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

T_p M,

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

J_p : Q_p \to Q_p

such that

J_p \circ J_p = - \operatorname_,

and an element

\xi_p

T_p M

which is not contained in

Q_p.

Given such data, one can define, for each

p

M,

a linear map

\eta_p : T_p M \to \R

and a linear map

\varphi_p : T_p M \to T_p M

\begin
\eta_p(u)&=0\textu\in Q_p\\
\eta_p(\xi_p)&=1\\
\varphi_p(u)&=J_p(u)\textu\in Q_p\\
\varphi_p(\xi)&=0.
\end

This defines a

one-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to e ...

\eta

and (1,1)-tensor field

\varphi

M,

and one can check directly, by decomposing

v

relative to the direct sum decomposition

T_p M = Q_p \oplus \left\,

that

\begin
\eta_p(v) \xi_p &= \varphi_p \circ \varphi_p(v) + v
\end

for any

v

T_p M.

Conversely, one may define an almost-contact structure as a triple

(\xi, \eta, \varphi)

which satisfies the two conditions *

\eta_p(v) \xi_p = \varphi_p \circ \varphi_p(v) + v

for any

v \in T_p M

\eta_p(\xi_p) = 1

Then one can define

Q_p

to be the

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...

of the linear map

\eta_p,

and one can check that the restriction of

\varphi_p

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. , * {{Manifolds Differential geometry Smooth manifolds