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Contact bundle

Bundle of linear subspaces of the tangent bundle

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Source updated October 17, 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 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.

01Definition

$M$ is an $n$ -dimensional smooth manifold. $TM$ is its tangent bundle. $T^{*}M$ is its cotangent bundle.

A contact element of order k at $p\in M$ is a $k$ plane $E\subset T_{p}M$ . For $k=n-1$ these are hyperplanes.

Given a vector space $V$ , the space of all k-dimensional subspaces of it is $\mathrm {Gr} _{k}(V)$ . It is the Grassmannian.

The $k$ -th contact bundle is the manifold of all order k contact elements: $C_{k}(M)=\bigsqcup _{p\in M}\mathrm {Gr} _{k}(T_{p}M)$ with the projection $\pi :C_{k}(M)\to M$ . This is a smooth fiber bundle with typical fiber $\mathrm {Gr} _{k}(\mathbb {R} ^{n})$ . For $1\leq k\leq n-1$ this produces $n-1$ distinct bundles. At each point of $M$ , the fiber is the space of all contact elements of order k through the point. $C_{k}(M)$ has dimension $n+(n-k)\times k$ .

$C_{k}(M)$ can also be constructed as an associated bundle of the frame bundle: $\operatorname {Fr} (TM)\times _{GL(n,\mathbb {R} )}\operatorname {Gr} _{k}\left(\mathbb {R} ^{n}\right)$ via the standard action of ${\textstyle GL(n,\mathbb {R} )$ on ${\textstyle \operatorname {Gr} _{k}\left(\mathbb {R} ^{n}\right)$ . The scalar subgroup ${\textstyle \mathbb {R} \times I_{n\times n}$ acts trivially, so its (effective) structure group is the projective linear group ${\textstyle PGL(n,\mathbb {R} )$ . Note that they are all associated with the same principal ${\textstyle GL(n,\mathbb {R} )$ -bundle.

02Examples

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

When $k=n-1$ , there is a natural identification with the projectivized cotangent bundle $\mathbb {P} (T^{*}M)$ . In this case the total space carries a natural contact structure induced by the tautological 1-form on $T^{*}M$ . In detail, a hyperplane $H\subset T_{p}M$ corresponds to a line of covectors in $T_{p}^{*}M$ , each of whose kernel is $H$ , giving $C_{n-1}(M)\cong \mathbb {P} (T^{*}M)$ . It is also called the bundle of hyperplane elements.

03Contact structure

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

Every k-dimensional submanifold of $M$ uniquely lifts to a k-dimensional submanifold of $C_{k}(M)$ . This is a generalization of the Gauss map. However, not every k-dimensional submanifold of $C_{k}(M)$ is a lift of a k-dimensional submanifold of $M$ . In fact, a k-dimensional submanifold of $C_{k}(M)$ is a lift of a k-dimensional submanifold of $M$ iff it is an integral manifold of a certain distribution in $C_{k}(M)$ . This distribution is called the contact structure of $C_{k}(M)$ .

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

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Sources and credits

This article is adapted from the Wikipedia article “Contact bundle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Contact bundle

01Definition

02Examples

03Contact structure

Sources and credits

Related topics

Grassmannian

Grassmann bundle

Jet bundle