Contact bundle
In differential geometry, a contact bundle is a particular type of fiber bundle constructed from a smooth manifold. Like how the tangent bundle is the manifold that describes the local behavior of parameterized curves, a contact bundle is the manifold that describes the local behavior of unparameterized curves. More generally, a contact bundle of order k is the manifold that describes the local behavior of k-dimensional submanifolds.
Since the contact bundle is obtained by combining Grassmannians of the tangent spaces at each point, it is a special case of the Grassmann bundle and of the projective bundle.
Definition
is an -dimensional smooth manifold. is its tangent bundle. is its cotangent bundle.A contact element of order k at is a plane. For these are hyperplanes.
Given a vector space, the space of all k-dimensional subspaces of it is. It is the Grassmannian.
The -th contact bundle is the manifold of all order k contact elements:with the projection. This is a smooth fiber bundle with typical fiber. For this produces distinct bundles. At each point of, the fiber is the space of all contact elements of order k through the point. has dimension.
can also be constructed as an associated bundle of the frame bundle:via the standard action of on. The scalar subgroup acts trivially, so its structure group is the projective linear group. Note that they are all associated with the same principal -bundle.
Examples
When, there is a canonical identification with the projectivized tangent bundle. It is also called the bundle of line elements. Each fiber is naturally identified with. If has a Riemannian metric, then its unit tangent bundle is a double cover of by forgetting the sign.When, there is a natural identification with the projectivized cotangent bundle. In this case the total space carries a natural contact structure induced by the tautological 1-form on. In detail, a hyperplane corresponds to a line of covectors in, each of whose kernel is, giving. It is also called the bundle of hyperplane elements.
Contact structure
Around each point of, construct local coordinate system. Each contact element then induces a local atlas of coordinate systems. The first system is of form, where is a matrix of shape. The others are obtained by permuting its columns.Every k-dimensional submanifold of uniquely lifts to a k-dimensional submanifold of. This is a generalization of the Gauss map. However, not every k-dimensional submanifold of is a lift of a k-dimensional submanifold of. In fact, a k-dimensional submanifold of is a lift of a k-dimensional submanifold of iff it is an integral manifold of a certain distribution in. This distribution is called the contact structure of.
In the special case where, the contact structure is a distribution of hyperplanes with dimension in the -dimensional manifold, and it is maximally non-integrable. In fact, "contact structure" usually refers to only distributions that are locally contactomorphic to this case of maximal non-integrability.