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 (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.

Definition

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

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

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

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

${\displaystyle C_{k}(M)}$ can also be constructed as an associated bundle of the frame bundle:$${\displaystyle \operatorname {Fr} (TM)\times _{GL(n,\mathbb {R} )}\operatorname {Gr} _{k}\left(\mathbb {R} ^{n}\right)}$$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.

Examples

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

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

Contact structure

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

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

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

See also

• Grassmannian
• Grassmann bundle
• Jet bundle
• Projectivization
• Contact geometry