O hiperbolični geometriji : delo diplomskega seminarja

Kastelec, Nika (author), Saksida, Pavle (mentor)

V nalogi spoznamo hiperbolično geometrijo prek treh modelov. Prvi obravnavani model je model zgornje polravnine ${\mathcal H}$, ki ga dobimo tako, da zgornjo polravnino opremimo s prvo fundamentalno formo psevdosfere, za katero smo dokazali,da ima konstantno negativno Gaussovo ukrivljenost. Ogledamo si, kaj so geodetke,ki jih imenujemo hiperbolične premice. Izpeljemo formulo za razdaljo na ${\mathcal H}$, $d(a, b) = 2 \tanh^{-1} {|b-a| \over |b-a|}$. Pokažemo, da so izometrije ${\mathcal H}$ translacije, vzporedne realni osi, zrcaljenja preko premic, vzporednih imaginarni osi, skaliranje za pozitiven realen faktor in zrcaljenje prek krožnice s središčem na realni osi ter končni kompozitumi naštetih preslikav. S preslikavo ${\mathcal P}(z)={z-i \over z+i}$ polravnino ${\mathcal H}$ preslikamo na enotski disk in ga opremili s tako prvo fundamentalno formo, da je preslikava ${\mathcal P}$ izometrija. S tem dobimo nov model hiperbolične geometrije imenovan Poincaréjev disk. Na tem modelu si ogledamo, kako izgleda vzporednost v hiperbolični geometriji, in ugotovili, da je smislno pojem vzporednosti razdeliti na dva pojma, vzporednost in ultra-vzporednost. Na koncu si ogledamo še nekoliko drugačen model, saj bo le ta vložen v ${\mathbb R}^3$ in ne v ravnino, kot prejšnja dva. Vzamemo enotsko sfero v metriki Minkowskega, ki je podana z matriko: $$J = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ Enotska sfera je v tem primeru dvodelni hiperboloid. Da bo opazovana ploskev povezana mnogoterost,opazujemo le zgornji del hiperboloida, ki ga označimo s ${\mathcal H}^2$. Ugotovimo, da metrika Minkowskega na tangentnem prostoru $T{\mathcal H}^2$ inducira pozitivno definitno kvadratno formo. Pokažemo, da obstaja izometrija med ${\mathcal H}^2$ in Poincaréjevem diskom. Na koncu klasificiramo izometrije hiperboloida ${\mathcal H}^2$. Ugotovili smo, da izometrije predstavlja grupa ${\rm SO}(2,1) = \{A \in {\rm GL}(3,{\mathbb R})|A^T JA = J\}$.

• Language: Slovenian
• Keywords: Hiperbolična geometrija, geodetske krivulje, izometrija, zgornja polravnina, Poincaréjev disk, hiperboloidni model, hiperbolična razdalja, psvdosfera
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-110792
• UDC: 514
• COBISS.SI-ID: 18820441
• Publication date in RUL: 20.09.2019

Secondary language

• Language: English
• Title: About hyperbolic geometry

Abstract

We present hyperbolic geometry through three different models. The first model is the upper half-plane model ${\mathcal H}$ where the half-plane is equiped with the first fundamental form of pseudosphere, for which we proved that it has constant negative Gaussian curvature. We explain what the geodetic curves are. These geodetic curves are called hyperbolic lines. We derive the formula of the distance on ${\mathcal H}$, $d(a, b) = 2 \tanh^{-1} {|b-a| \over |b-a|}$. We show that the isometries of ${\mathcal H}$ are translation parallel to the real axis, reflections through lines parallel to the imaginary axis, dilations by factor $a \in R$, inversions in circles with centres on the real axis and compositions of finite number of the mentioned maps. Half-plane ${\mathcal H}$ was maped with ${\mathcal P}(z)={z-i \over z+i}$ to the unit disc. The disc was equiped with the first fundamental form such that the map ${\mathcal P}$ is isometry. So we get the second model of the hyperbolic geometry called Poincaré's disc. On this model we present two kind of hyperbolic parallels: parallels and ultra-parallels. The third model of the hyperbolic geometry is contrary to the first two, included in ${\mathbb R}^3$. The model is based on the unit sphere of Minkowsky's metric defined by matrix: $$J = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ The unit sphere is in this case the double hyperboloid. To obtain a connected manifold we took only the upper part of it and we denotes it by ${\mathcal H}^2$. We found out that the Minkovsky's matrix on the tangent space $T{\mathcal H}^2$ induces positive definite quadratic form. We show that the isometry between ${\mathcal H}^2$ and Poincaré's disc exists. We show that the isometry of ${\mathcal H}^2$ is the group ${\rm SO}(2,1) = \{A \in {\rm GL}(3,{\mathbb R})|A^T JA = J\}$.

• Keywords: Hyperbolic geometry, geodetic curves, isometry, upper half-plane, Poincar é's disc, hyperboloid model, hyperbolic distance, pseudosphere