Kontaktna geometrija

magistrsko delo

Andrej Svetina (Author), Franc Forstnerič (Mentor)

Abstract

V delu se ukvarjamo s kontaktnimi strukturami na gladkih mnogoterostih, tj. popolnoma neintegrabilnimi podsvežnji kodimenzije ena v tangentnem svežnju dane mnogoterosti. Obravnavamo Grayev izrek, ki pravi, da so članice gladke družine kontaktnih struktur na sklenjeni mnogoterosti med seboj kontaktomorfne. Dokažemo Darbouxov izrek o lokalni ekvivalenci kontaktnih struktur na mnogoterostih iste razsežnosti. Slednji je poseben primer izreka o okolicah sklenjenih izotropnih podmnogoterosti v dani ambientni mnogoterosti. Difeomorfni izotropni podmnogoterosti v različnih mnogoterostih imata namreč kontaktomorfni cevasti okolici, čim sta njuna konformna simplektična normalna svežnja izomorfna. Pogoj je trivialno izpolnjen za Legendrove podmnogoterosti, kar napelje na podrobnejšo obravnavo krivulj in vozlov v tri-mnogoterostih. Pokažemo, da lahko poljubno zvezno krivuljo aproksimiramo z Legendrovo v C0-topologiji. Dokažemo tudi obstoj kontaktnih struktur na sklenjenih in orientabilnih trirazsežnih mnogoterostih. Definiramo pojem zavite kontaktne strukture na trirazsežni mnogoterosti in obravnavamo njene osnovne lastnosti.

Keywords

kontaktna geometrija;Legendrove krivulje;zavite kontaktne strukture;

Data

Language:	Slovenian
Year of publishing:	2020
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[A. Svetina]
UDC:	514.7
COBISS:	27532547
Views:	824
Downloads:	176
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Contact geometry
Secondary abstract:	In this thesis we study contact structures on smooth manifolds, i. e. completely nonintegrable subbundles of codimension one of the tangent bundle. We prove Gray's theorem, which states that the members of a smooth family of contact structures on a closed manifold are pairwise contactomorphic. Darboux's theorem, which we also prove, states that contact manifolds have no local invariants, apart from dimension. It is itself a special case of the isotropic neighbourhood theorem, which allows us to identify tubular neighbourhoods of diffeomorphic isotropic submanifolds in two different contact manifolds, provided they have isomorphic conformal symplectic normal bundles. This condition holds trivially for Legendrian submanifolds, which in turn allows a more detailed study of curves and knots in three-manifolds. We show that any continuous curve can be approximated with a Legendrian one in C0-topology. We prove that closed orientable three manifolds always admit contact structures. We define overtwisted contact structures on three-manifolds and study their basic properties.
Secondary keywords:	contact geometry;Legendrian curves;overtwisted contact structures;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 2. stopnja
Pages:	IX, 71 str.
ID:	12027604