{"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."}],"ddc":["510"],"user_id":"85279","oa":"1","citation":{"mla":"Offen, Christian. “Local Intersections of Lagrangian Manifolds Correspond to Catastrophe  Theory.” <i>ArXiv:1811.10165</i>.","apa":"Offen, C. (n.d.). Local intersections of Lagrangian manifolds correspond to catastrophe  theory. In <i>arXiv:1811.10165</i>.","ieee":"C. Offen, “Local intersections of Lagrangian manifolds correspond to catastrophe  theory,” <i>arXiv:1811.10165</i>. .","short":"C. Offen, ArXiv:1811.10165 (n.d.).","ama":"Offen C. Local intersections of Lagrangian manifolds correspond to catastrophe  theory. <i>arXiv:181110165</i>.","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.” <i>ArXiv:1811.10165</i>, n.d."},"file":[{"creator":"coffen","file_name":"LocalLagrangianContact.pdf","description":"Two smooth map germs are right-equivalent if and only if they\ngenerate two Lagrangian submanifolds in a cotangent bundle which have the\nsame contact with the zero-section. In this paper we provide a reverse direction\nto this classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a symplectic manifold have the same contact with a third Lagrangian\nsubmanifold if and only if the intersection problems correspond to stably right\nequivalent map germs. We, therefore, obtain a correspondence between local\nLagrangian intersection problems and catastrophe theory while the classical\nversion only captures tangential intersections. The correspondence is defined\nindependently of any Lagrangian fibration of the ambient symplectic manifold, in contrast to other classical results. Moreover, we provide an extension\nof the correspondence to families of local Lagrangian intersection problems.\nThis gives rise to a framework which allows a natural transportation of the\nnotions of catastrophe theory such as stability, unfolding and (uni-)versality\nto the geometric setting such that we obtain a classification of families of lo-\ncal Lagrangian intersection problems. An application is the classification of\nLagrangian boundary value problems for symplectic maps.","file_size":662483,"date_updated":"2021-12-21T16:01:03Z","file_id":"29078","access_level":"open_access","title":"Local intersections of Lagrangian manifolds correspond to catastrophe theory","date_created":"2021-12-21T16:01:03Z","content_type":"application/pdf","relation":"main_file"}],"department":[{"_id":"636"}],"title":"Local intersections of Lagrangian manifolds correspond to catastrophe  theory","external_id":{"arxiv":["1811.10165"]},"date_created":"2020-10-06T16:32:45Z","publication":"arXiv:1811.10165","file_date_updated":"2021-12-21T16:01:03Z","type":"preprint","year":"2018","_id":"19940","language":[{"iso":"eng"}],"status":"public","has_accepted_license":"1","author":[{"full_name":"Offen, Christian","last_name":"Offen","orcid":"https://orcid.org/0000-0002-5940-8057","id":"85279","first_name":"Christian"}],"date_updated":"2023-08-10T08:48:55Z","publication_status":"submitted"}