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master's thesis
Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru
Zagreb: University of Zagreb, Faculty of Science, 2021. urn:nbn:hr:217:350949
Martinović, Filip
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Martinović, F. (2021). Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:350949

Martinović, Filip. "Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:350949

Martinović, Filip. "Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru." Master's thesis, University of Zagreb, Faculty of Science, 2021. https://urn.nsk.hr/urn:nbn:hr:217:350949

Martinović, F. (2021). 'Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru', Master's thesis, University of Zagreb, Faculty of Science, accessed 21 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:350949

Martinović F. Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2021 [cited 2024 December 21] Available at: https://urn.nsk.hr/urn:nbn:hr:217:350949

F. Martinović, "Vremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2021. Available at: https://urn.nsk.hr/urn:nbn:hr:217:350949

Metadata
TitleVremenske i svjetlosne plohe u Lorentz-Minkowskijevom prostoru
Title (english)Time and light surfaces in Lorentz-Minkowski space
AuthorFilip Martinović
MentorŽeljka Milin Šipuš (mentor)
Committee memberŽeljka Milin Šipuš (predsjednik povjerenstva)
Committee memberDražen Adamović (član povjerenstva)
Committee memberMarko Vrdoljak (član povjerenstva)
Committee memberMarko Erceg (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2021-07-22, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Cilj ovog rada jest obraditi primjere vremenskih i svjetlosnih ploha Lorentz-Minkowskijevog prostora \(\mathbb{R}^3_1\). Da bismo proučili te primjere, potrebno je razraditi diferencijalnu geometriju trodimenzionalnog realnog prostora s nedegeneriranom metrikom indeksa 1. U prvom poglavlju ispitujemo kakvu geometriju inducira ta metrika na sam prostor \(\mathbb{R}^3_1\), klasifciramo vektore i potprostore tog prostora te ispitujemo svojstva ortogonalnosti. Zatim, u drugom poglavlju ispitujemo svojstva krivulja tog prostora, defniramo njihovu Freneteovu bazu, izvodimo Frenet-Serreteove formule te dokazujemo analogone fundamentalnog teorema krivulja. U trećom poglavlju proučavamo plohe prostora \(\mathbb{R}^3_1\), klasifciramo ih na prostorne, vremenske i svjetlosne te proučavamo kakvu geometriju na njima inducira metrika prostora \(\mathbb{R}^3_1\). Konačno, koristeći razrađenu teoriju, ispitujemo primjere vremenskih i svjetlosnih ploha, dakle ploha čija je inducirana metrika nedegenrirana indefnitna odnosno degenerirana i time nalazimo primjere ploha koje nemaju analogon u klasičnoj euklidskoj geometriji.
Abstract (english)
Goal of this paper is to examine examples of timelike and lightlike surfaces in Lorentz- Minkowski space \(\mathbb{R}^3_1\). To do so, we need to study differential geometry of the threedimensional real space with nondegenerate metric of index 1. In first chapter we inspect geometric properties that this metric induces on space \(\mathbb{R}^3_1\), we classify vectors and subspaces of this space and we investigate properties of orthogonality. Next, in second chapter we explore properties of curves in this space, we define corresponding Frenet frame, we derive Frenet-Serret equations and we prove theorems analogous to standard fundamental theorems for curves. In third chapter we study surfaces of \(\mathbb{R}^3_1\), we classify them as spacelike, timelike and lightlike and we examine what kind of geometry does the metric of \(\mathbb{R}^3_1\) induces on the surfaces. Lastly, by using developed theory we inspect examples of timelike and lightlike surfaces, that is surfaces with induced nondegenerate indefinite metric or degenerate metric and with that we find examples of surfaces that have no analogues in standard Euclidean geometry.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:350949
Study programmeTitle: Pure Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2021-09-16 09:22:05