Made available in DSpace on 2015-05-14T12:14:07Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1123884 bytes, checksum: 0b9e6a7e26799ac54ef0dcb5fb98d15b (MD5) Previous issue date: 2012-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We consider the geometry of affine connections and take, as particular examples, Weyl and Riemann-Cartan geometies. In a modern geometrical approach, we take up the problem of local embedding of manifolds in Weyl spaces and in spaces endowed with semi-symmetric torsion. We then obtain the extrinsic curvature, Weingarten operator and Gauss-Codazzi equations in the mentioned non-riemannian spaces. We investigate some important properties of a Weyl structure in the case of a warped product and carry out an analysis of the geodesics in a foliation de…ned in such a space. We consider the particular case when the embedding space is a warped product manifold and has a Riemann-Cartan geometry. As an application, we show that the torsion …eld of de bulk may provide a mechanism of geometrical con…nement. In this way, we exhibit a classical analogue of the quantum con…nement induced by scalar …elds. / Consideramos a geometria de uma conexão a…m e abordamos como exemplos, as geometrias de Weyl e Riemann-Cartan, esta ultima considerando o caso em que a torção é semi-simétrica. Após uma exposição moderna das propriedades destas geometrias, abordamos o problema de imersões isométricas em espaços de Weyl e de torção semi-simétrica. Introduzimos um roteiro para a obtenção da curvatura extrínseca, operador de Weingarten e das equações de Gauss-Codazzi para tais espaços. Em seguida, analisamos as propriedades de uma estrutura de Weyl em um espaço produto distorcido (EPD) e analisamos as geodésicas das folhas em tal espaço. Consideramos, também, o caso particular quando o espaço ambiente para um (EPD) com uma geometria de Riemann-Cartan. Mostramos como o confi…namento e as propriedades de estabilidade de geodésicas próximas ao mundo-brana podem ser afetadas pela torção do bulk. Deste modo, construímos um análogo clássico do confi…namento quântico inspirado em modelos de teoria de campo, substituindo um campo escalar por um campo de torção.

Bien qu’ayant vu le jour dans un cadre dit relativiste avec l’avènement de la théorie de la relativité générale, le lien intime existant entre géométrie de l’espace-temps d’une part, et gravitation d’autre part, peut se voir étendu aux théories dites nonrelativistes, l’exemple paradigmatique en étant la reformulation géométrique de la gravitation Newtonienne initiée par E. Cartan. De tels espace-temps nonrelativistes diffèrent structurellement de leurs homologues relativistes, ces disparités étant le plus naturellement expliquées en réinterprétant ces premiers comme réduction dimensionnelle d’espace-temps relativistes privilégiés. L’ambition de cette thèse est double : Dans une première partie, nous nous intéressons à une généralisation de la classe d’espace-temps relativistes permettant le formalisme ambiant, étudions leur interprétation géométrique ainsi que la classe élargie de structures nonrelativistes pouvant y être plongées. La seconde partie de ce manuscrit concerne le point de vue, informé par la théorie des groupes, que porte E. Cartan sur la géométrie différentielle et plus précisément l’éclairage que projettent les géométries de Cartan sur les structures nonrelativistes, à la fois dans leur définition intrinsèque et dans leur relation avec des structures relativistes au travers du formalisme ambiant. / With the advent of general relativity, the profound interaction between the geometry of spacetime and gravitational phenomena became a truism of modern physics. However, the intimate relationship between spacetime geometry and gravitation is by no means restricted to relativistic physics but can in fact be successfully applied to nonrelativistic physics, the paradigmatic example being E. Cartan geometrisation of Newtonian gravity. This geometrisation of nonrelativistic gravitation involves some nonrelativistic structures whose discrepancies in comparison with their relativistic peers are better understood when embedded inside specific classes of relativistic gravitational waves. The ambition of this Doctoral Thesis is twofold: In a first part, we discuss a generalisation of the class of gravitational waves allowing the embedding of nonrelativistic features, explore their geometric properties and the new nonrelativistic structures emerging from this study. In a second part, we advocate how the group-theoretically oriented approach of Cartan to differential geometry can shed new light on nonrelativistic structures, both in an intrinsic and ambient fashion.

Submitted by HENDRIK GERARD JOHAN JENNEN null (hjennen@ift.unesp.br) on 2016-02-23T14:54:31Z No. of bitstreams: 1 thesis_oneside.pdf: 1083742 bytes, checksum: eeb3f42f2937a777dba99b7615ef69c8 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-02-24T13:39:03Z (GMT) No. of bitstreams: 1 jannen_h_dr_ift.pdf: 1083742 bytes, checksum: eeb3f42f2937a777dba99b7615ef69c8 (MD5) / Made available in DSpace on 2016-02-24T13:39:03Z (GMT). No. of bitstreams: 1 jannen_h_dr_ift.pdf: 1083742 bytes, checksum: eeb3f42f2937a777dba99b7615ef69c8 (MD5) Previous issue date: 2016-02-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Observações realizadas nas últimas três décadas confirmaram que o universo se encontra em um estado de expansão acelerada. Essa aceleração é atribuída à presença da chamada energia escura, cuja origem permanece desconhecida. A maneira mais simples de se modelar a energia escura consiste em introduzir uma constante cosmológica positiva nas equações de Einstein, cuja solução no vácuo é então dada pelo espaço de de Sitter. Isso, por sua vez, indica que a cinemática subjacente ao espaço-tempo deve ser aproximadamente governada pelo grupo de de Sitter SO(1,4), e não pelo grupo de Poincaré ISO(1,3). Nesta tese, adotamos tal argumento como base para a conjectura de que o grupo que governa a cinemática local é o grupo de de Sitter, com o desvio em relação ao grupo de Poincaré dependendo ponto-a-ponto do valor de um termo cosmológico variável. Com o propósito de desenvolver tal formalismo, estudamos a geometria de Cartan na qual o espaço modelo de Klein é, em cada ponto, um espaço de de Sitter com o conjunto de pseudo-raios definindo uma função não-constante do espaço-tempo. Encontramos que o tensor de torção nessa geometria adquire uma contribuição que não está presente no caso de uma constante cosmológica. Fazendo uso da teoria das realizações não-lineares, estendemos a classe de simetrias do grupo de Lorentz SO(1,3) para o grupo de de Sitter. Em seguida, verificamos que a estrutura da gravitação teleparalela--- uma teoria gravitacional equivalente à relatividade geral--- é uma geometria de Riemann-Cartan não linear. Inspirados nesse resultado, construímos uma generalização da gravitação teleparalela sobre uma geometria de de Sitter--Cartan com um termo cosmológico dado por uma função do espaço-tempo, a qual é consistente com uma cinemática localmente governada pelo grupo de de Sitter. A função cosmológica possui sua própria dinâmica e emerge naturalmente acoplada não-minimalmente ao campo gravitacional, analogamente ao que ocorre nos modelos telaparalelos de energia escura ou em teorias de gravitação escalares-tensoriais. Característica peculiar do modelo aqui desenvolvido, a função cosmológica fornece uma contribuição para o desvio geodésico de partículas adjacentes em queda livre. Embora tendo sua própria dinâmica, a energia escura manifesta-se como um efeito da cinemática local do espaço-tempo. / Observations during the last three decades have confirmed thatthe universe momentarily expands at an accelerated rate, which is assumed to be driven by dark energy whose origin remains unknown. The minimal manner of modelling dark energy is to include a positive cosmological constant in Einstein's equations, whose solution in vacuum is de Sitter space. This indicates that the large-scale kinematics of spacetime is approximated by the de Sitter group SO(1,4) rather than the Poincaré group ISO(1,3). In this thesis we take this consideration to heart and conjecture that the group governing the local kinematics of physics is the de Sitter group, so that the amount to which it is a deformation of the Poincaré group depends pointwise on the value of a nonconstant cosmological function. With the objective of constructing such a framework we study the Cartan geometry in which the model Klein space is at each point a de Sitter space for which the combined set of pseudoradii forms a nonconstant function on spacetime. We find that the torsion receives a contribution that is not present for a cosmological constant. Invoking the theory of nonlinear realizations we extend the class of symmetries from the Lorentz group SO(1,3) to the enclosing de Sitter group. Subsequently, we find that the geometric structure of teleparallel gravity--- a description for the gravitational interaction physically equivalent to general relativity--- is a nonlinear Riemann--Cartan geometry.This finally inspires us to build on top of a de Sitter--Cartan geometry with a cosmological function a generalization of teleparallel gravity that is consistent with a kinematics locally regulated by the de Sitter group. The cosmological function is given its own dynamics and naturally emerges nonminimally coupled to the gravitational field in a manner akin to teleparallel dark energy models or scalar-tensor theories in general relativity. New in the theory here presented, the cosmological function gives rise to a kinematic contribution in the deviation equation for the world lines of adjacent free-falling particles. While having its own dynamics, dark energy manifests itself in the local kinematics of spacetime.

In dieser Dissertation werden Cartan-Ränder von CR-Mannigfaltigkeiten und ihren Fefferman-Räumen besprochen. Der Fefferman-Raum einer strikt pseudo-konvexen CR-Mannigfaltigkeit ist als das Bündel aller reellen Strahlen im kanonischen, komplexen Linienbündel definiert. Eine andere Definition nutzt die Cartan-Geometrie und führt zu einer starken Beziehung zwischen den Cartan-Geometrien der CR-Mannigfaltigkeit und des zugehörigen Fefferman-Raumes. Allerdings wird hier die Existenz einer gewissen Wurzel des antikanonischen, komplexen Linienbündels, dessen Existenz nur lokal gesichert ist, vorausgesetzt. Für Randkonstruktionen benötigen wir jedoch eine globale Konstruktion des Fefferman-Raumes. Dennoch können lokale Resultate zum Fefferman-Raum von einer Konstruktion zur anderen übertragen werden können, da konforme Überlagerungen von beiden vorliegen. Der Cartan-Rand einer Mannigfaltigkeit wird mithilfe der zugehörigen Cartan-Geometrie konstruiert, welche eine globale Basis und damit auch eine Riemannsche Metrik auf dem Cartan-Bündel definiert, welches per Cauchy-Vervollständigung abgeschlossen wird. Division durch die Strukturgruppe ergibt den Cartan-Rand der Mannigfaltigkeit. Der Cartan-Rand ist eine Verallgemeinerung des Cauchy-Randes, da beide im Riemannschen übereinstimmen. Allgemein ist der Cartan-Rand nicht unbedingt Hausdorffsch, was nicht wirklich überrascht, sind doch Rand-Phänomene "irgendwie singulär". Wir stellen fest, dass für CR-Mannigfaltigkeit und ihre Fefferman-Räume die Projektion des Cartan-Randes des Fefferman-Raumes den Cartan-Rand der CR-Mannigfaltigkeit enthält. Schließlich betrachten wir die Heisenberg-Gruppe, eines der grundlegenden Beispiele für CR-Mannigfaltigkeiten. Sie ist flach aber - anders als der homogene Raum - nicht kompakt. Wir finden, dass der Cartan-Rand der Heisenberg-Gruppe ein einzelner Punkt und der Cartan-Rand des zugehörigen Fefferman-Raumes eine nicht-ausgeartete Faser über diesem ist. / The aim of this thesis is to discuss the Cartan boundaries of CR manifolds and their Fefferman spaces. The Fefferman space of a strictly pseudo-convex CR manifold is defined as the bundle of all real rays in the canonical complex line bundle. Another way of defining the Fefferman space of a CR manifold uses the tools of Cartan geometry and leads to a strong relationship between the Cartan geometries of a CR manifold and the corresponding Fefferman space. However here the existence of a certain root of the anticanonical complex line bundle is requested which can solely be guarantied locally. As we are interested in boundaries we need a global construction of the Fefferman space. Still we find that local results on the Fefferman space can be transferred from one construction to the other since we have conformal coverings of both. The Cartan boundary of a manifold is constructed with the help of the corresponding Cartan geometry, which defines a global frame and hence a Riemannian metric on the Cartan bundle which can be completed by Cauchy completion. Division by the structure group gives the Cartan boundary of the manifold. The Cartan boundary is a generalization of the Cauchy boundary since both coincide in the Riemannian case. In general the Cartan boundary is not necessarily Hausdorff, which is not really surprising since boundary phenomena are somehow ``singular''''. For CR manifolds and their Fefferman spaces we especially prove that the projection of the Cartan boundary of the Fefferman space contains the Cartan boundary of the CR manifold. We finally discuss the Heisenberg group, one of the basic examples of CR manifolds. It is flat but - contrary to the homogeneous space - not compact. We find that the Cartan boundary of the Heisenberg group is a single point and the Cartan boundary of the corresponding Fefferman space is a non degenerate fibre over that point.