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am starting an entry contact manifold
Any more idea about whether contact geometry is likely to be a general feature of geometry?
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.
at contact manifold I have added the explicit statement of the Boothby-Wang theorem.
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.
Hi Zoran,
not sure what the intent of the comment is. But please feel invited to add discussion to contact manifold!
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