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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2012

    am starting an entry contact manifold

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 27th 2012

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

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2012
    • (edited Jun 27th 2012)

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

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

    One can then form the automorphism group

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

    in the slice \infty-topos H /BU(1)\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 AA on the total space PP of the circle bundle PXP \to X. Then (P,A)(P,A) is a regular contact structure and the above automorphisms in the slice \infty-topos are identified with the diffeomorphisms PPP \to P that preserves AA, 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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2013

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

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeApr 14th 2013
    • (edited Apr 14th 2013)

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2013

    Hi Zoran,

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

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2020

    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

  1. Added references

    Rongmin Lu

    diff, v16, current