Contact bundle

Short description: Bundle of linear subspaces of the tangent 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

$M$ 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 ${\mathrm{G}\mathrm{r}}_k(V)$. It is the Grassmannian.

The $k$-th contact bundle is the manifold of all order k contact elements:$$C_k(M)=\underset{p\in M}{⨆}{\mathrm{G}\mathrm{r}}_k(T_{p}M)$$with the projection $\pi :C_k(M)\to M$. This is a smooth fiber bundle with typical fiber ${\mathrm{G}\mathrm{r}}_k({\mathbb{ℝ}}^n)$. For $1\le k\le n-1$ 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:$$\mathrm{Fr}(TM){\times }_{GL(n,\mathbb{ℝ})}{\mathrm{Gr}}_k\left({\mathbb{ℝ}}^n\right)$$via the standard action of $GL(n,\mathbb{ℝ})$ on ${\mathrm{Gr}}_k\left({\mathbb{ℝ}}^n\right)$. The scalar subgroup $\mathbb{ℝ}\times I_{n\times n}$ acts trivially, so its (effective) structure group is the projective linear group $PGL(n,\mathbb{ℝ})$. Note that they are all associated with the same principal $GL(n,\mathbb{ℝ})$-bundle.

Examples

When $k=1$, there is a canonical identification with the projectivized tangent bundle $\mathbb{\mathbb{P}}(TM)$. It is also called the bundle of line elements. Each fiber ${\mathrm{G}\mathrm{r}}_1({\mathbb{ℝ}}^n)$ is naturally identified with ${\mathbb{ℝ}\mathbb{\mathbb{P}}}^{n-1}$. 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 $\mathbb{\mathbb{P}}(T^{*}M)$. 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 $C_{n-1}(M)\cong \mathbb{\mathbb{P}}(T^{*}M)$. It is also called the bundle of hyperplane elements.

Contact structure

Around each point of $M$, construct local coordinate system $q^1,\dots ,q^n$. Each contact element then induces a local atlas of $(\genfrac{}{}{0ex}{}{n}{k})$ coordinate systems. The first system is of form $\left[\begin{array}{c}{I}_{(n-k)\times (n-k)}|A\end{array}\right]$, 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.

See also

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

References

• Blair, David E. (2010) (in en). Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics. 203 (2nd ed.). Boston, MA: Birkhäuser. doi:10.1007/978-0-8176-4959-3. ISBN 978-0-8176-4958-6. https://link.springer.com/book/10.1007/978-0-8176-4959-3. 
• Burke, William L. (1985). Applied differential geometry (Reprint ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-26317-7.