Almost-contact manifold


In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a smooth manifold, obtained by combining a contact-element structure and an almost-complex structure. They can be considered as an odd-dimensional counterpart to almost complex manifolds.
They were introduced by John Gray in 1959. Shigeo Sasaki in 1960 introduced Sasakian manifold to study them.

Definition

Given a smooth manifold, an almost-contact structure is a triple 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 contact element, a linear complex structure on it, and an element of which is not contained in. As usual, the selection must be smooth.
Equivalently, one may define an almost-contact structure as a triple, where is a vector field on, is a 1-form on, and is a (1,1)-tensor field on, such that they satisfy the two conditionsOr in more detail, for any and any,
Because the choice of the transverse vector field is smooth, the field is a co-orientation of the distribution of contact elements.
More abstractly, it can be defined as a G-structure obtained by reduction of the structure group from to.

Equivalence

In one direction, given, one can define for each in a linear map and a linear map byand one can check directly, by decomposing relative to the direct sum decomposition, thatfor any in.
In another direction, given, 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.

Properties

Given an almost contact structure on a -manifold, we have:
  • has rank 2n.

Relation to other manifolds

Metric

Given an almost-contact manifold equipped with the previously defined, we may add a Riemannian metric to it. We say the metric is compatible with the almost-contact structure iff the metric satisfies the metric compatibility condition:Such a manifold is called an almost contact metric manifold.
Define the fundamental 2-form by. Then is skew-symmetric and.
Compatible metrics are easy to find. That is, they are not rigid. To construct one, take any metric, and let, then this is a compatible metric:Special cases used in the literature are:
  • Contact metric manifold: additionally and.
  • Sasakian manifold: contact metric manifold, with normality condition.
  • Almost coKähler manifold: almost contact metric, with and .

Classification

They have been fully classified via group representation theory into 4096 classes.
Let be an almost contact metric structure on a -manifold, and let. At each point, regardwhere For, it splits into orthogonal, irreducible, -invariant subspacesAn almost contact metric manifold is of class if for all. Hence there are classes.
Given such a manifold, it can be classified as follows: compute, project it onto the twelve , and identify the class by which components are nonzero.
Specific cases named in the literature:
  • Cosymplectic:.
  • Nearly -cosymplectic:.
  • Almost cosymplectic:.
  • -Kenmotsu:.
  • -Sasakian:.
  • Trans-Sasakian.
  • Quasi-Sasakian:.

Examples

A cosymplectic structure on a smooth manifold of dimension induces an almost-contact structure. Specifically, a cosymplectic structure is a tuple where is a closed 1-form, is a closed 2-form, and at every point. One way to produce a cosymplectic structure is by foliating the manifold into symplectic manifolds, and set to be the symplectic structure on each manifold, and have parallel to the tangent planes through the foliation.
Another common way to construct a cosymplectic structure is through time-dependent Hamiltonian mechanics. Let a phase space be. A trajectory of a system in phase space is a path in. Let be canonical coordinates on the phase space, which may be allowed to vary over time. Then provides is an almost-contact structure on the manifold.
The construction of the almost-contact metric structure:
To show it, note thatThus on all of. Hence is an almost-contact structure.