Neste trabalho estudamos alguns resultados importantes sobre a teoria das subvariedades bi-harmônicas de espaços homogêneos tridimensionais. Existem três classes de espaços homogêneos tridimensionais simplesmente conexos dependendo da dimensão do grupo de isometrias, que pode ser: 3, 4 ou 6. No caso da dimensão ser 6, M é uma forma espacial; se a dimensão do grupo de isometrias for 4, M é isométrica a: \'H IND. 3\' (grupo de Heisenberg), SU(2) (grupo unitário especial), ~SL(2,R) (revestimento universal do grupo linear especial), ou aos espaços produtos \'S POT. 2\' × R e \'H POT. 2\' × R. Feita exceção para \'H POT. 3\', no caso da dimensão ser 4 ou 6 o espaço homogêneo é localmente isométrico a (uma parte de) \'R POT. 3\', munido de uma métrica que depende de dois parâmetros reais. Tal família de métricas aparece primeiramente no trabalho [3] de L. Bianchi e, mais tarde, nos artigos [14, 35] de É. Cartan e G. Vranceanu, respectivamente. Nesse projeto de mestrado, queremos estudar (essencialmente) resultados de existência e classificação de subvariedades bi-harmônicas nesses espaços, também conhecidos como variedades de Bianchi-Cartan-Vranceanu / In this work we study some important results about the theory of the biharmonic submanifolds of tridimensional homogeneous spaces. There exist three classes of simply connected tridimensional homogeneous spaces depending on the dimension of the group of isometries, which can be: 3, 4 or 6. In the case of dimension 6, M will be a space form; if the dimension of the group of isometries is 4, M will be isometric to: either \'H IND. 3\' (Heisenbergs group), or SU(2) (special unitary group), or ~SL(2,R) (universal recovering of the special linear group), or the product spaces \'S POT. 2\' × R and \'H POT. 2\' × R. Except for \'H POT. 3\', in the case of dimension 4 or 6 the homogeneous space is locally isometric to (a part of) \'R POT. 3\', endowed with a metric that depends on two real parameters. Such family of metrics first appears in the work [3] of L. Bianchi and later in the articles [14, 35] of ´E. Cartan and G. Vranceanu, respectively. In this master thesis, we want to study (essentially) results of existence and classification of bi-harmonic submanifolds in these spaces, also known as Bianchi-Cartan-Vranceanus manifolds

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2008. / Submitted by Jaqueline Oliveira (jaqueoliveiram@gmail.com) on 2008-12-03T16:50:58Z No. of bitstreams: 1 DISSERTACAO_2008_IgorSantosLima.pdf: 492821 bytes, checksum: f9908ee8af40bc5f62f504e6f76b0cb8 (MD5) / Approved for entry into archive by Georgia Fernandes(georgia@bce.unb.br) on 2009-02-03T17:17:04Z (GMT) No. of bitstreams: 1 DISSERTACAO_2008_IgorSantosLima.pdf: 492821 bytes, checksum: f9908ee8af40bc5f62f504e6f76b0cb8 (MD5) / Made available in DSpace on 2009-02-03T17:17:04Z (GMT). No. of bitstreams: 1 DISSERTACAO_2008_IgorSantosLima.pdf: 492821 bytes, checksum: f9908ee8af40bc5f62f504e6f76b0cb8 (MD5) / Os grupos de Bianchi Γd = PSL2(Od) com d > 0 inteiro livre de quadrados, onde Od é o anel de inteiros do corpo Q(p√(-d)), são importantes devido as suas aplicações em Geometria e Teoria dos Números. A propriedade de separabilidade sob conjugação é importante pois foi notado por Mal´cev em 1958 que o problema da conjugação para grupos finitamente apresentados separáveis sob conjugação é solúvel. Em 1998, Wilson e Zalesski demonstraram que Γd é separável sob conjugação para d = 1; 2; 7; 11 e conjecturaram que todos Γd são separáveis sob conjugação. Nesta tese foi provado que 3 é separável sob conjugação. Combinando com o resultado acima, fica mostrado que todos os grupos de Bianchi Euclideanos são separáveis sob conjugação. ______________________________________________________________________________________________ ABSTRACT / TThe Bianchi groups Γd = PSL2(Od) with d > 0 square-free integer, where Od is the ring of integers of the field Q(p√(-d)), are important because their applications in Geometry and Number Theory. The conjugacy separability property is important because as noted by Mal´cev in 1958, the conjugacy problem for finitely presented conjugacy separable groups is soluble. In 1998, Wilson and Zalesski showed that d is conjugacy separable for d = 1; 2; 7; 11 and they conjectured that all Γd are conjugacy separable. In the thesis was proved that Γ3 is conjugacy separable. Combining with previous results this shows that all Euclidean Bianchi groups are conjugacy separable.

In this work the methods of mode decomposition and Fourier analysis of quantum fields on curved spacetimes previously available mainly for the scalar fields on Friedman-Robertson-Walker spacetimes are extended to arbitrary vector fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier analysis. Explicit constructions are performed for a variety of situations arising in homogeneous cosmology. A number of results concerning classical and quantum fields known for very restricted situations are generalized to cover almost all cosmological models.

Made available in DSpace on 2017-07-21T14:42:56Z (GMT). No. of bitstreams: 1 Francieli Lunelli Santos.pdf: 9973901 bytes, checksum: 0a6c5b596d9885ddd8a7083e7d6e18da (MD5) Previous issue date: 2009-07-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The picture of family groups, as well as other groups, is one way to spread Imaging speeches and emphasized social relations that were frozen at the moment there portrayed. The practice of families of using the photographer to get group pictures remained for much of the 20th century in the city of Ponta Grossa. It is understood that the families that put a pose for the photographer constructed an idealized image of the group for the picture. Performed thus a representation, a social production, which consolidated the way the family would be recognized. Thus, studying the social representations of family portraits in the period 1915-1940, means building not only possible interpretations for the family, but to social groups in general, in that period. Based on the issues mentioned above, this dissertation aimed to reconstruct meanings and practices in social representations related to the act of photographing families, in Ponta Grossa, in the period cited. This work is divided into three chapters. The first deals with the theoretical concepts that give support to work. They are: family, social representations and social roles. In the second chapter sets up the city, the Bianchi´s family and the installation of their photo studio, where it gave rise to images of families. The final chapter is devoted to the analysis of iconography and iconology pictures, its constituent elements and symbols that pervade the representations of their characters. / A fotografia de grupos familiares, assim como de outros grupos, constitui-se em um meio imagético para se disseminar discursos e ressaltar relações sociais que ali ficaram congeladas no instante retratado. A prática das famílias recorrerem ao fotógrafo para obterem retratos em grupos manteve-se durante boa parte do século XX na cidade de Ponta Grossa. Entende-se que as famílias que se colocavam a posar para o fotógrafo construíam uma imagem idealizada do grupo para o retrato. Realizava, dessa forma, uma representação, uma encenação social, que consolidava a maneira como a família gostaria de ser reconhecida. Assim, estudar as representações sociais nos retratos de família do período compreendido entre 1915-1940 significa construir interpretações possíveis não apenas para a família, mas para os grupos sociais de maneira geral, no referido período. Com base nas questões acima apontadas, a presente dissertação objetivou reconstruir significados e práticas nas representações sociais ligadas ao ato de fotografar famílias, em Ponta Grossa, no período citado. Este trabalho está dividido em três capítulos. No primeiro abordam-se os conceitos teóricos que dão sustentação ao trabalho. São eles: família, representações sociais e papéis sociais. No segundo capítulo apresenta-se a cidade, a família Bianchi e a instalação de seu estúdio fotográfico, onde se deu origem às imagens de famílias. O último capítulo destina-se à análise iconográfica e iconológica dos retratos, seus elementos constitutivos e símbolos que permeiam as representações dos seus personagens.

In this work the methods of mode decomposition and Fourier analysis of quantum fields on curved spacetimes previously available mainly for the scalar fields on Friedman-Robertson-Walker spacetimes are extended to arbitrary vector fields on general spatially homogeneous spacetimes. This is done by developing a rigorous unified framework which incorporates mode decomposition, harmonic analysis and Fourier analysis. Explicit constructions are performed for a variety of situations arising in homogeneous cosmology. A number of results concerning classical and quantum fields known for very restricted situations are generalized to cover almost all cosmological models.

La grande précision des mesures du fond diffus cosmologique nécessitent de comprendre avec finesse la physique sous-jacente afin d'en tirer des conclusions pertinentes sur la phase primordiale de l'univers. Dans cette thèse nous étudions la théorie des perturbations non-linéaires dans le cadre de la relativité générale. Notre but est de déterminer le transfert des perturbations de la métrique ainsi que des perturbations du contenu matériel,<br />entre la phase primordiale de l'univers et les observations réalisées<br />aujourd'hui. Nous nous plaçons tout d'abord dans l'approximation fluide afin d'appréhender les comportements généraux attendus. Ensuite nous étudions la théorie cinétique au second ordre, nécessaire pour obtenir le transfert radiatif non-linéaire, dans le but de déterminer la non-gaussianité dans le<br />fond diffus cosmologique. <br />Nous étudions également la théorie des perturbations linéaires<br />autour d'espaces anisotropes. Nous élaborons la théorie des perturbations invariantes de jauge autour d'un espace de Bianchi I, puis nous étudions les signatures observationnelles d'une phase primordiale d'inflation possédant cette symétrie.

This is a thesis on general relativity. It analyzes dynamical properties of Einstein's field equations in cosmology and in the vicinity of spacetime singularities in a number of different situations. Different techniques are used depending on the particular problem under study; dynamical systems methods are applied to cosmological models with spatial homogeneity; Hamiltonian methods are used in connection with dynamical systems to find global monotone quantities determining the asymptotic states; Fuchsian methods are used to quantify the structure of singularities in spacetimes without symmetries. All these separate methods of analysis provide insights about different facets of the structure of the equations, while at the same time they show the relationships between those facets when the different methods are used to analyze overlapping areas. The thesis consists of two parts. Part I reviews the areas of mathematics and cosmology necessary to understand the material in part II, which consists of five papers. The first two of those papers uses dynamical systems methods to analyze the simplest possible homogeneous model with two tilted perfect fluids with a linear equation of state. The third paper investigates the past asymptotic dynamics of barotropic multi-fluid models that approach a `silent and local' space-like singularity to the past. The fourth paper uses Hamiltonian methods to derive new monotone functions for the tilted Bianchi type II model that can be used to completely characterize the future asymptotic states globally. The last paper proves that there exists a full set of solutions to Einstein's field equations coupled to an ultra-stiff perfect fluid that has an initial singularity that is very much like the singularity in Friedman models in a precisely defined way. / <p>Status of the paper "Perfect Fluids and Generic Spacelike Singularities" has changed from manuscript to published since the thesis defense.</p>

The goal of this thesis is two-fold. First, it gives an efficient method for calculating the action of Hecke operators in terms of "Manin" symbols, otherwise known as "M-symbols," in the first homology group of Bianchi spaces. Second, it presents data that may be used to understand and better state an unpublished conjecture of Fukaya, Kato, and Sharifi concerning the structure of Bianchi Spaces modulo Eisenstein ideals [5]. Swan, Cremona, and others have studied the homology of Bianchi spaces characterized as certain quotients of hyperbolic 3-space [3], [13]. The first homology groups are generated both by modular symbols and a certain subset of them: the Manin symbols. This is completely analogous to the study of the homology of modular curves. For modular curves, Merel developed a technique for calculating the action of Hecke operators completely in terms of "Manin" symbols [10]. For Bianchi spaces, Bygott and Lingham outlined methods for calculating the action of Hecke operators in terms of modular symbols [2], [9]. This thesis generalizes the work of Merel to Bianchi spaces. The relevant Bianchi spaces are characterized by imaginary quadratic fields K. The methods described in this thesis deal primarily with the case that the ring of integers of K is a PID. Let p be an odd prime that is split in K. The calculations give the F_p-dimension of the homology modulo both p and an Eisenstein ideal. Data is given for primes less than 50 and the five Euclidean imaginary quadratic fields Q(√-1), Q(√-2), Q(√-3), Q(√-7), and Q(√-11). All of the data presented in this thesis comes from computations done using the computer algebra package Magma.

No abstract available. / Thesis (M.Sc.)-University of Natal, 1997.

Le modèle standard de la cosmologie est fondé sur les hypothèses d'homogénéité et d'isotropie de l'Univers. Pour interpréter la plupart des observations, ces deux hypothèses sont appliquées de façon stricte ; l’objectif principal de cette thèse a été d'évaluer leur pertinence, en particulier lorsque de très petites échelles sont mises en jeu. Après une revue détaillée des lois de l'optique géométrique en espace-temps courbe, on propose une analyse exhaustive de la propagation de la lumière à travers des modèles cosmologiques « en gruyère », modélisant le caractère grumeleux de l'Univers. L'impact sur l'interprétation du diagramme de Hubble s'avère être faible, en particulier grâce à la constante cosmologique. Lorsqu'appliquées aux données actuelles issues de l'observation de supernovae, les corrections associées tendent à améliorer l'accord entre les paramètre cosmologiques mesurés à partir du diagramme de Hubble et du fond diffus cosmologique. Ceci suggère que la précision des observations cosmologiques atteinte aujourd'hui ne permet plus de négliger l'effet des petites structures sur la propagation de lumière à travers le cosmos. Un tel constat a motivé le développement d'un nouveau cadre théorique, inspiré de la physique statistique, visant à décrire ces effets avec précision. Quant à l'hypothèse d'isotropie, cette thèse aborde d'une part les conséquences potentielles d'une anisotropie cosmique sur la propagation de la lumière, en résolvant les équation de l'optique géométrique dans l'espace-temps de Bianchi I. D'autre part, on y analyse une classe de sources d'anisotropie, à savoir les modèles scalaire-vecteur. / The standard model of cosmology is based on the hypothesis that the Universe is spatially homogeneous and isotropic. When interpreting most observations, this cosmological principle is applied stricto sensu ; the main goal of the present thesis was to evaluate how reliable this assumption is, especially when small scales are at stake. After having reviewed the laws of geometric optics in curved spacetime, and the standard interpretation of cosmological observables, the dissertation reports a comprehensive analysis of light propagation in Swiss-cheese models, designed to capture the clumpy character of the Universe. The resulting impact on the interpretation of the Hubble diagram is quantified, and shown to be relatively small, thanks to the cosmological constant. When applied to current supernova data, the associated corrections tend to improve the agreement between the cosmological parameters inferred from the Hubble diagram and from the cosmic microwave background. This is a hint that the effect of small-scale structures on light propagation may become non-negligible in the era of precision cosmology. This motivated the development of a new theoretical framework, based on stochastic processes, which aims at describing small-scale lensing with a better accuracy. Regarding the isotropy side of the cosmological principle, this dissertation addresses, on the one hand, the potential effect of a large-scale anisotropy on light propagation, by solving the equations of geometric optics in the Bianchi I spacetime. On the other hand, possible sources of such an anisotropy, namely scalar-vector models for inflation or dark energy, are analysed.