Contact bundle

{{Short description|Bundle of linear subspaces of the tangent bundle}} {{No footnotes|date=October 2025}} 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 (of order 1) is the manifold that describes the local behavior of ''un''parameterized 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 == <math>M</math> is an <math>n</math>-dimensional smooth manifold. <math>TM</math> is its tangent bundle. <math>T^* M</math> is its cotangent bundle.

A '''contact element''' of order ''k'' at <math>p\in M</math> is a <math>k</math> plane <math>E\subset T_pM</math>. For <math>k=n-1</math> these are hyperplanes.

Given a vector space <math>V</math>, the space of all ''k''-dimensional subspaces of it is <math>\mathrm{Gr}_k(V)</math>. It is the '''Grassmannian'''.

The <math>k</math>-th '''contact bundle''' is the manifold of all order ''k'' contact elements:<math display="block">C_k(M)=\bigsqcup_{p\in M}\mathrm{Gr}_k(T_pM)</math>with the projection <math>\pi:C_k(M)\to M</math>. This is a smooth fiber bundle with typical fiber <math>\mathrm{Gr}_k(\mathbb{R}^n)</math>. For <math>1\le k\le n-1</math> this produces <math>n-1</math> distinct bundles. At each point of <math>M</math>, the fiber is the space of all contact elements of order ''k'' through the point. <math>C_k(M)</math> has dimension <math>n + (n-k) \times k</math>.

<math>C_k(M)</math> can also be constructed as an associated bundle of the frame bundle:<math display="block">\operatorname{Fr}(T M) \times_{G L(n, \mathbb{R})} \operatorname{Gr}_k\left(\mathbb{R}^n\right)</math>via the standard action of <math display="inline">G L(n, \mathbb{R})</math> on <math display="inline">\operatorname{Gr}_k\left(\mathbb{R}^n\right)</math>. The scalar subgroup <math display="inline">\mathbb{R} \times I_{n \times n}</math> acts trivially, so its (effective) structure group is the projective linear group <math display="inline">P G L(n, \mathbb{R})</math>. Note that they are all associated with the same principal <math display="inline">G L(n, \mathbb{R})</math>-bundle.

== Examples == When <math>k=1 </math>, there is a canonical identification with the projectivized tangent bundle <math>\mathbb{P}(TM)</math>. It is also called the '''bundle of line elements'''. Each fiber <math>\mathrm{Gr}_1(\mathbb{R}^n)</math> is naturally identified with <math>\mathbb{RP}^{\,n-1}</math>. If <math>M</math> has a Riemannian metric, then its unit tangent bundle <math>UT(M) </math> is a double cover of <math>C_1(M)</math> by forgetting the sign.

When <math>k=n-1</math>, there is a natural identification with the projectivized cotangent bundle <math>\mathbb{P}(T^*M)</math>. In this case the total space carries a natural contact structure induced by the tautological 1-form on <math>T^*M</math>. In detail, a hyperplane <math>H\subset T_pM</math> corresponds to a line of covectors in <math>T_p^*M</math>, each of whose kernel is <math>H</math>, giving <math>C_{n-1}(M)\cong \mathbb{P}(T^*M)</math>. It is also called the '''bundle of hyperplane elements'''.

== Contact structure == Around each point of <math>M</math>, construct local coordinate system <math>q^1, \dots, q^n</math>. Each contact element then induces a local atlas of <math>\binom{n}{k}</math> coordinate systems. The first system is of form <math>\begin{bmatrix} I_{(n-k)\times(n-k)} | A \end{bmatrix}</math>, where <math>A</math> is a matrix of shape <math>(n-k) \times k</math>. The others are obtained by permuting its columns.

Every ''k''-dimensional submanifold of <math>M</math> uniquely lifts to a ''k''-dimensional submanifold of <math>C_k(M)</math>. This is a generalization of the Gauss map. However, not every ''k''-dimensional submanifold of <math>C_k(M)</math> is a lift of a ''k''-dimensional submanifold of <math>M</math>. In fact, a ''k''-dimensional submanifold of <math>C_k(M)</math> is a lift of a ''k''-dimensional submanifold of <math>M</math> iff it is an integral manifold of a certain distribution in <math>C_k(M)</math>. This distribution is called the '''contact structure''' of <math>C_k(M)</math>.

In the special case where <math>k = n-1</math>, the contact structure is a distribution of hyperplanes with dimension <math>(2n-2)</math> in the <math>(2n-1)</math>-dimensional manifold <math>C_{n-1}(M)</math>, 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.

== References == {{reflist}} {{reflist|group=note}}

* {{cite book | last = Blair | first = David E. | title = Riemannian Geometry of Contact and Symplectic Manifolds | edition = 2nd | date = 2010 | publisher = Birkhäuser | location = Boston, MA | series = Progress in Mathematics | volume = 203 | isbn = 978-0-8176-4958-6 | doi = 10.1007/978-0-8176-4959-3 | url = https://link.springer.com/book/10.1007/978-0-8176-4959-3 | language = en }} * {{Cite book |last=Burke |first=William L. |title=Applied differential geometry |date=1985 |publisher=Cambridge University Press |isbn=978-0-521-26317-7 |edition=Reprint |location=Cambridge}}

Category:Differential geometry Category:Smooth manifolds Category:Fiber bundles