Local intersections of Lagrangian manifolds correspond to catastrophe theory
C. Offen, ArXiv:1811.10165 (n.d.).
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Abstract
Two smooth map germs are right-equivalent if and only if they generate two
Lagrangian submanifolds in a cotangent bundle which have the same contact with
the zero-section. In this paper we provide a reverse direction to this
classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a
symplectic manifold have the same contact with a third Lagrangian submanifold
if and only if the intersection problems correspond to stably right equivalent
map germs. We, therefore, obtain a correspondence between local Lagrangian
intersection problems and catastrophe theory while the classical version only
captures tangential intersections. The correspondence is defined independently
of any Lagrangian fibration of the ambient symplectic manifold, in contrast to
other classical results. Moreover, we provide an extension of the
correspondence to families of local Lagrangian intersection problems. This
gives rise to a framework which allows a natural transportation of the notions
of catastrophe theory such as stability, unfolding and (uni-)versality to the
geometric setting such that we obtain a classification of families of local
Lagrangian intersection problems. An application is the classification of
Lagrangian boundary value problems for symplectic maps.
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arXiv:1811.10165
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Cite this
Offen C. Local intersections of Lagrangian manifolds correspond to catastrophe theory. arXiv:181110165.
Offen, C. (n.d.). Local intersections of Lagrangian manifolds correspond to catastrophe theory. In arXiv:1811.10165.
@article{Offen, title={Local intersections of Lagrangian manifolds correspond to catastrophe theory}, journal={arXiv:1811.10165}, author={Offen, Christian} }
Offen, Christian. “Local Intersections of Lagrangian Manifolds Correspond to Catastrophe Theory.” ArXiv:1811.10165, n.d.
C. Offen, “Local intersections of Lagrangian manifolds correspond to catastrophe theory,” arXiv:1811.10165. .
Offen, Christian. “Local Intersections of Lagrangian Manifolds Correspond to Catastrophe Theory.” ArXiv:1811.10165.
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Local intersections of Lagrangian manifolds correspond to catastrophe theory
Description
Two smooth map germs are right-equivalent if and only if they
generate two Lagrangian submanifolds in a cotangent bundle which have the
same contact with the zero-section. In this paper we provide a reverse direction
to this classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a symplectic manifold have the same contact with a third Lagrangian
submanifold if and only if the intersection problems correspond to stably right
equivalent map germs. We, therefore, obtain a correspondence between local
Lagrangian intersection problems and catastrophe theory while the classical
version only captures tangential intersections. The correspondence is defined
independently of any Lagrangian fibration of the ambient symplectic manifold, in contrast to other classical results. Moreover, we provide an extension
of the correspondence to families of local Lagrangian intersection problems.
This gives rise to a framework which allows a natural transportation of the
notions of catastrophe theory such as stability, unfolding and (uni-)versality
to the geometric setting such that we obtain a classification of families of lo-
cal Lagrangian intersection problems. An application is the classification of
Lagrangian boundary value problems for symplectic maps.
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2021-12-21T16:01:03Z
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