198 research outputs found
Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions
There is a remarkable type of field of two-planes special to four dimensions
known as an Engel distributions. They are the only stable regular distributions
besides the contact, quasi-contact and line fields. If an arbitrary two-plane
field on a four-manifold is slightly perturbed then it will be Engel at generic
points. On the other hand, if a manifold admits an oriented Engel structure
then the manifold must be parallelizable and consequently the alleged Engel
distribution must have a degeneration loci -- a point set where the Engel
conditions fails. By a theorem of Zhitomirskii this locus is a finite union of
surfaces. We prove that these surfaces represent Chern classes associated to
the distribution.Comment: LaTeX, 15 page
Cotangent models for integrable systems
We associate cotangent models to a neighbourhood of a Liouville torus in
symplectic and Poisson manifolds focusing on a special class called
-Poisson/-symplectic manifolds. The semilocal equivalence with such
models uses the corresponding action-angle coordinate theorems in these
settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds
and an action-angle theorem for regular Liouville tori in Poisson manifolds
[LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but
also consider the Liouville tori on the singular locus of a -Poisson
manifold. For this latter class of Poisson structures we define a twisted
cotangent model. The equivalence with this twisted cotangent model is given by
an action-angle theorem recently proved in [KMS16]. This viewpoint of cotangent
models provides a new machinery to construct examples of integrable systems,
which are especially valuable in the -symplectic case where not many sources
of examples are known. At the end of the paper we introduce non-degenerate
singularities as lifted cotangent models on -symplectic manifolds and
discuss some generalizations of these models to general Poisson manifolds.Comment: 25 pages; final version to appear at Communications in Mathematical
Physic
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