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.
Definition
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.
Examples
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.
Contact structure
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.