Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 27th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexam starting an entry _[[contact manifold]]_

   am starting an entry contact manifold

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 27th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAny more idea about whether contact geometry is likely to be a [general feature of geometry](http://golem.ph.utexas.edu/category/2012/02/prequantization_in_homotopy_ty.html#c040901)?

   Any more idea about whether contact geometry is likely to be a general feature of geometry?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 27th 2012
   • (edited Jun 27th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, that's why I made that note: in as far as we are dealing with _regular_ contact manifolds, they are precisely nothing but the total spaces of circle bundles with connection. So here is one way to look at it from general abstraction: in $\mathbf{H} := Smooth\infty Grpd$ a circle bundle with connection on some $X$ is given by a morphism $$ \nabla: X \to \mathbf{B}U(1)_{conn} \,. $$ One can then form the automorphism group $$ \mathbf{Aut}(\nabla) $$ in the slice $\infty$-topos $\mathbf{H}_{/\mathbf{B}U(1)}$. This, is, as discussed in [that entry](http://golem.ph.utexas.edu/category/2012/02/prequantization_in_homotopy_ty.html#c040901) that you point to, the [[quantomorphism group]] of $\nabla$ regarded as a [[prequantum circle bundle]]. Contact geometry provides one presentation of this computation: we incarnate $\nabla$ equivalently as an [[Ehresmann connection]] 1-fom $A$ on the total space $P$ of the [[circle bundle]] $P \to X$. Then $(P,A)$ is a regular contact structure and the above automorphisms in the slice $\infty$-topos are identified with the diffeomorphisms $P \to P$ that preserves $A$, hence with those contactomorphisms that are, again, quantomorphisms. In summary: circle bundles with connection are a "first principles" notion in cohesive homotopy type theory, and _regular_ contact geometry is one way of looking at them, regarded as objects in the slice, as above. I am not sure yet if there is anything along these lines to be said about the non-regular case of contact geometry. I might just say "it's an approximation to the regular case when a prequantum circle bundle does not quite exist". But maybe there is more to it.

   Yes, that’s why I made that note:

   in as far as we are dealing with regular contact manifolds, they are precisely nothing but the total spaces of circle bundles with connection.

   So here is one way to look at it from general abstraction:

   in $\mathbf{H} := Smooth\infty Grpd$ a circle bundle with connection on some $X$ is given by a morphism

   $$\nabla: X \to \mathbf{B}U(1)_{conn} \,.$$

   One can then form the automorphism group

   $$\mathbf{Aut}(\nabla)$$

   in the slice $\infty$-topos $\mathbf{H}_{/\mathbf{B}U(1)}$. This, is, as discussed in that entry that you point to, the quantomorphism group of $\nabla$ regarded as a prequantum circle bundle.

   Contact geometry provides one presentation of this computation: we incarnate $\nabla$ equivalently as an Ehresmann connection 1-fom $A$ on the total space $P$ of the circle bundle $P \to X$. Then $(P,A)$ is a regular contact structure and the above automorphisms in the slice $\infty$-topos are identified with the diffeomorphisms $P \to P$ that preserves $A$, hence with those contactomorphisms that are, again, quantomorphisms.

   In summary: circle bundles with connection are a “first principles” notion in cohesive homotopy type theory, and regular contact geometry is one way of looking at them, regarded as objects in the slice, as above.

   I am not sure yet if there is anything along these lines to be said about the non-regular case of contact geometry. I might just say “it’s an approximation to the regular case when a prequantum circle bundle does not quite exist”. But maybe there is more to it.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 13th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexat _[[contact manifold]]_ I have added the explicit statement of the Boothby-Wang theorem.

   at contact manifold I have added the explicit statement of the Boothby-Wang theorem.

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeApr 14th 2013
   • (edited Apr 14th 2013)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn Arnold's book __Mathematical methods of classical mechacnics__, one of the appendices is dedicated to contact manifolds. Arnol'd explains that the __main idea__ is that the contact manifolds are maximally away from an integrability proprerty; the 1-form is introduced precisely to express this idea. His exposition is crisp, but not that easy.

   In Arnold’s book Mathematical methods of classical mechacnics, one of the appendices is dedicated to contact manifolds. Arnol’d explains that the main idea is that the contact manifolds are maximally away from an integrability proprerty; the 1-form is introduced precisely to express this idea. His exposition is crisp, but not that easy.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi Zoran, not sure what the intent of the comment is. But please feel invited to add discussion to _[[contact manifold]]_!

   Hi Zoran,

   not sure what the intent of the comment is. But please feel invited to add discussion to contact manifold!

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 18th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * Alfonso Giuseppe Tortorella, Luca Vitagliano & Ori Yudilevich, _Homogeneous G-structures_, Annali di Matematica (2020) ([doi:10.1007/s10231-020-00972-9](https://doi.org/10.1007/s10231-020-00972-9)) <a href="https://ncatlab.org/nlab/revision/diff/contact+manifold/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/contact+manifold/15">v15</a>, <a href="https://ncatlab.org/nlab/show/contact+manifold">current</a>

   added pointer to:

   • Alfonso Giuseppe Tortorella, Luca Vitagliano & Ori Yudilevich, Homogeneous G-structures, Annali di Matematica (2020) (doi:10.1007/s10231-020-00972-9)

   diff, v15, current

8. • CommentRowNumber8.
   • CommentAuthornLab edit announcer
   • CommentTimeAug 10th 2021
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexAdded references Rongmin Lu <a href="https://ncatlab.org/nlab/revision/diff/contact+manifold/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/contact+manifold/16">v16</a>, <a href="https://ncatlab.org/nlab/show/contact+manifold">current</a>

   Added references

   Rongmin Lu

   diff, v16, current