---
_id: '19940'
abstract:
- lang: eng
text: "Two smooth map germs are right-equivalent if and only if they generate two\r\nLagrangian
submanifolds in a cotangent bundle which have the same contact with\r\nthe zero-section.
In this paper we provide a reverse direction to this\r\nclassical result of Golubitsky
and Guillemin. Two Lagrangian submanifolds of a\r\nsymplectic manifold have the
same contact with a third Lagrangian submanifold\r\nif and only if the intersection
problems correspond to stably right equivalent\r\nmap germs. We, therefore, obtain
a correspondence between local Lagrangian\r\nintersection problems and catastrophe
theory while the classical version only\r\ncaptures tangential intersections.
The correspondence is defined independently\r\nof any Lagrangian fibration of
the ambient symplectic manifold, in contrast to\r\nother classical results. Moreover,
we provide an extension of the\r\ncorrespondence to families of local Lagrangian
intersection problems. This\r\ngives rise to a framework which allows a natural
transportation of the notions\r\nof catastrophe theory such as stability, unfolding
and (uni-)versality to the\r\ngeometric setting such that we obtain a classification
of families of local\r\nLagrangian intersection problems. An application is the
classification of\r\nLagrangian boundary value problems for symplectic maps."
accept: '1'
author:
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: https://orcid.org/0000-0002-5940-8057
citation:
ama: Offen C. Local intersections of Lagrangian manifolds correspond to catastrophe
theory. *arXiv:181110165*.
apa: Offen, C. (n.d.). Local intersections of Lagrangian manifolds correspond to
catastrophe theory. *ArXiv:1811.10165*.
bibtex: '@article{Offen, title={Local intersections of Lagrangian manifolds correspond
to catastrophe theory}, journal={arXiv:1811.10165}, author={Offen, Christian}
}'
chicago: Offen, Christian. “Local Intersections of Lagrangian Manifolds Correspond
to Catastrophe Theory.” *ArXiv:1811.10165*, n.d.
ieee: C. Offen, “Local intersections of Lagrangian manifolds correspond to catastrophe
theory,” *arXiv:1811.10165*. .
mla: Offen, Christian. “Local Intersections of Lagrangian Manifolds Correspond to
Catastrophe Theory.” *ArXiv:1811.10165*.
short: C. Offen, ArXiv:1811.10165 (n.d.).
date_created: 2020-10-06T16:32:45Z
date_updated: 2021-02-07T13:37:11Z
ddc:
- '510'
department:
- _id: '636'
file:
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2021-02-07T13:35:22Z
date_updated: 2021-02-07T13:35:22Z
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.\nIn this paper we provide a reverse direction to this classical
result of Golubitsky and Guillemin.\nTwo 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.\nThe
correspondence is defined independently of any Lagrangian fibration of the ambient
symplectic manifold, in contrast to other classical results. \nMoreover, we provide
an extension of the correspondence to families of local Lagrangian intersection
problems.\nThis 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.\nAn application is the classification of Lagrangian
boundary value problems for symplectic maps."
file_id: '21186'
file_name: LocalLagrangianContact.pdf
file_size: 536316
relation: main_file
request_a_copy: 0
title: Local intersections of Lagrangian manifolds correspond to catastrophe theory
file_date_updated: 2021-02-07T13:35:22Z
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1811.10165
oa: 1
publication: arXiv:1811.10165
publication_status: submitted
status: public
title: Local intersections of Lagrangian manifolds correspond to catastrophe theory
type: preprint
user_id: '85279'
year: '2018'
...