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Humboldt-Universiẗat, Diss., 2001--Berlin.

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.

Cette thèse a pour objet principal l'étude des structures pseudo-riemanniennes et de leurs groupes de transformations conformes, locales et globales. On cherche à obtenir des informations générales sur la structure du groupe conforme d'une variété pseudo-riemannienne compacte de dimension au moins 3, et on s'intéresse également à la géométrie et la dynamique des actions conformes de groupes de Lie sur de telles structures. L'essentiel des résultats présentés en géométrie conforme se situe en signature lorentzienne (1,n-1).Le point de vue qui est adopté ici est d'interpréter une structure conforme de dimension au moins 3 comme étant la donnée d'une géométrie de Cartan modelée sur l'univers d'Einstein de même signature. Ces structures géométriques, introduites par Élie Cartan, sont rigides et leurs symétries locales ont des propriétés remarquables. Nous retrouvons dans ce contexte des résultats formulés par Mikhaïl Gromov à la fin des années 1980, et les mettons en œuvre sur le cas particulier de la géométrie de Cartan définie par une structure conforme. / The main object of this thesis is the study of pseudo-Riemannian structures and their local and global conformal transformation groups. The purpose is to obtain general informations about the conformal group of a compact pseudo-Riemannian manifold of dimension greater than or equal to 3, and we also study dynamical and geometrical properties of conformal Lie group actions on such structures. The largest part of the result that are presented in this work are formulated in the (1,n-1) Lorentz signature.The approach we have chosen here to study a conformal structure is to work with its associated normal Cartan geometry modeled on the Einstein universe with same signature. These geometric structures, introduced by Élie Cartan, are rigid and their local automorphisms have nice behaviours. We formulate in this context results of Mikhaïl Gromov, that go back to the late 1980', and use them in the particular case of the normal Cartan geometry associated to a conformal structure.

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

Nesse trabalho, estudamos o método do referencial móvel e sistemas diferenciais exteriores. Estabelecemos resultados de Geometria Riemanniana via referenciais móveis e com essa linguagem introduzimos o Teorema de Gauss-Bonnet-Chern e apresentamos uma adaptação da demonstração original de S.-S. Chern presente no artigo A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ao abordar aspectos da teoria de Cartan-Kähler, codificamos as ideias oriundas dos referenciais móveis em sistemas diferenciais exteriores e mostramos algumas aplicações à Geometria Riemanniana. / In this work, we study the method of moving frame and exterior differential systems. We set up results of Riemannian Geometry via moving frames and with this language we introduce the Gauss-Bonnet-Chern Theorem and present an adaptation of the original proof of S.-S. Chern in the article A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. In discussing aspects of Cartan-Kählers theory, we encode the ideas from moving frames into exterior differential systems and use this tool in Riemannian Geometry.

One deals in this work with the existence and the uniqueness of natural projectively equivariant quantizations by means of the theory of Cartan connections. One shows that a natural projectively equivariant quantization exists for differential operators acting between $lambda$ and $mu$-densities if and only if the corresponding $sl(m+1,mathbb{R})$-equivariant quantization on $mathbb{R}^{m}$ exists. With this end in view, one writes the quantization by means of a formula in terms of the normal Cartan connection associated to the projective structure of a connection. One deduces next an explicit formula for the natural projectively equivariant quantization. One shows the non-uniqueness of such a quantization by means of the curvature of the normal Cartan connection. Finally, one shows the existence of natural and projectively equivariant quantizations for differential operators acting between sections of other natural fiber bundles transposing the method used in $mathbb{R}^{m}$ to analyse the existence of $sl(m+1,mathbb{R})$-equivariant quantizations, this method being linked to the Casimir operator./ On traite dans cet ouvrage de l'existence et de l'unicité de quantifications naturelles projectivement équivariantes au moyen de la théorie des connexions de Cartan. On démontre qu'une quantification naturelle projectivement équivariante existe pour des opérateurs différentiels agissant entre $lambda$ et $mu$-densités si et seulement si la quantification $sl(m+1,mathbb{R})$- équivariante correspondante sur $mathbb{R}^{m}$ existe. Pour cela, on exprime la quantification au moyen d'une formule en termes de la connexion de Cartan normale associée à la structure projective d'une connexion. On en déduit ensuite une formule explicite pour la quantification naturelle projectivement invariante. On démontre après la non-unicité d'une telle quantification par le biais de la courbure de la connexion de Cartan normale. Enfin, on démontre l'existence de quantifications naturelles projectivement équivariantes pour des opérateurs différentiels agissant entre sections d'autres fibrés naturels en transposant la méthode utilisée dans $mathbb{R}^{m}$ pour analyser l'existence de quantifications $sl(m+1,mathbb{R})$-équivariantes, méthode liée à l'opérateur de Casimir.

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.

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