Preslikavanja pravčastih ploha u Minkowskijevom prostoru

doctoral thesis

• Title (english): Mappings of ruled surfaces in Minkowski space
• Author: Ljiljana Primorac Gajčić
• Mentor: Željka Milin-Šipuš (mentor)
• Committee member: Dragutin Svrtan (predsjednik povjerenstva)
• Committee member: Blaženka Divjak (član povjerenstva)
• Committee member: Zdenka Kolar-Begović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-10-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U ovoj disertaciji se proučavaju preslikavanja pravčastih ploha u Minkowskijevom prostoru, specijalnom ambijentalnom prostoru u kojemu, s obzirom na definiranu pseudometriku, razlikujemo tri vrste vektora, krivulja, odnosno ploha. Pravčaste plohe su plohe koje dozvoljavaju parametrizaciju oblika \(f(u, v)=c(u)+ve(u)\), gdje su \(u, v\) realni brojevi. Krivulja \(c\) se naziva bazna krivulja, a pravci određeni vektorom smjera \(e(u)\) nazivaju se izvodnice. U uvodnom dijelu disertacije dan je pregled osnovnih pojmova i teorema za Minkowskijev prostor, odnosno lokalna teorija ploha u Minkowskijevom prostoru, te njihova klasifikacija s obzirom na kauzalni karakter bazne krivulje, odnosno izvodnica pravčaste plohe. U Poglavlju 2 proučavane su izometrije pravčastih ploha, te su posebno analizirani uvjeti kada navedeno preslikavanje čuva izvodnice pravčastih ploha, tzv. Mindingove izometrije. Ovakvo preslikavanje je proučavano za 3 relevantne klase pravčastih ploha koje se javljaju u Minkowskijevom prostoru. Posebna pažnja je posvećena plohama koje nemaju svoj analogon u euklidskom prostoru, te je uveden pojam nul pravčaste plohe konstantnog nagiba i dana njihova karakterizacija. U Poglavljima 3, odnosno 4 proučavana su konformna, odnosno ekviarealna preslikavanja za sve klase pravčastih ploha, ponovno uz dodatni uvjet da navedena preslikavanja čuvaju izvodnice. U Poglavlju 5 proučavane su invarijante Mindingove izometrije pravčastih ploha u \(n\)-dimenzionalnom Minkowskijevom prostoru, \(\mathbb{R}_1^n\).

Abstract (english)

In this dissertation we study mappings of ruled surfaces in Minkowski space, special ambient space in which, with respect to defined pseudo-metrics, we distinguish three types of vectors, curves and surfaces. Ruled surfaces are surfaces that admit a parameterization of the form \(f(u, v)=c(u) +ve(u)\), where \(u, v\) are real numbers. The curve \(c\) is called the base curve and the straight lines determined by direction vectors \(e(u)\) are called the rulings of a ruled surface. In Preliminaries of the thesis we give an overview of basic concepts and theorems of Minkowski space, local theory of surfaces in Minkowski space, as well as their classification with respect to the causal character of the base curve, respectively rulings of ruled surface. In Chapter 2 we studied isometries of ruled surfaces, with particular interest in analyzing the conditions when given mapping preserves rulings of ruled surfaces, so-called Minding isometry. This mapping is studied for three relevant classes of ruled surfaces that appear in the Minkowski space. Special attention was given to surfaces that have no Euclidean counterparts, and we introduced the concept of null-ruled surface of constant slope and gave theorems for their characterizations. In Chapters 3, respectively 4 we studied conformal, respectively area-preserving mappings for every class of ruled surfaces, also with additional condition that those mappings preserve rulings of ruled surface. In Chapter 5 we studied the invariants of Minding isometries of ruled surfaces in \(n\)-dimensional Minkowski space, \(\mathbb{R}_1^n\)

• Keywords: Minkowskijev prostor; pravčaste plohe; lokalne izometrije; Mindingove izometrije; konformna preslikavanja; ekviarealna preslikavanja; (k+1)-pravčaste plohe
• Keywords (english): Minkowski space; ruled surfaces; local isometry; Minding isometry; conformal mapping; areapreserving mapping; (k+1)-ruled surfaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:647354
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: vii, 93 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-21 11:21:13