Contact bundle

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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.

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

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

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

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

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

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

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

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

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

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

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

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