Characterizing a manifold up to isometry is a challenging task. A manifold is a topological space. One may equip a manifold with a metric, and generally speaking, this metric determines how the manifold “looks". An example of this would be the unit sphere in R3. While we typically envision the standard metric on this sphere to give it its familiar shape, one could define a different metric on this set of points, distorting distances within this set to make it seem perhaps more ellipsoidal, something not isometric to the standard round sphere. In an effort to distinguish manifolds up to isometry, we wish to compute meaningful invariants. For example, the Riemann curvature tensor and its surrogates are examples of invariants one could construct. Since these objects are generally too complicated to compare and are not real valued, we construct scalar invariants from these objects instead. This thesis will explore these invariants and exhibit a special family of manifolds that are not flat on which all of these invariants vanish. We will go on to properly define, and gives examples of, manifolds, metrics, tangent vector fields, and connections. We will show how to compute the Christoffel symbols that define the Levi-Civita connection, how to compute curvature, and how to raise and lower indices so that we can produce scalar invariants. In order to construct the curvature operator and curvature tensor, we use the miracle of pseudo-Riemannian geometry, i.e., the Levi-Civita connection, the unique torsion free and metric compatible connection on a manifold. Finally, we examine Generalized Plane Wave Manifolds, and show that all scalar invariants of Weyl type on these manifolds vanish, despite the fact that many of these manifolds are not flat.

Einstein's equations in vacuum spacetimes with two spacelike killing vectors are explored using affine projection tensor geometry. By doing a semi-conformal transformation on the metric, a new "fiducial" geometry is constructed using a projection tensor fields. This fiducial geometry provides coordinate independent information about the underlying structure of the spacetime without the use of an explicit form of the metric tensor.

La déformation visco-plastique de la glace est fortement anisotrope, le plan de glissement préferé étant la plan de base. Le fait que dans un polycristal chaque grain possède sa propre direction de déformation produit des incompatibilités et un champ de contrainte complexe. La déformation à été étudiée expérimentellement en mésurant la dis- tortion des plans cristallins de mono- et polycristaux de glace artificielle. Les expériences ont été réalisées à l'aide d'un faisceau synchrotron. Une nouvelle procédure éxperimental, basée sur les méthodes de l'imagerie en diffraction de Bragg, comme lumière blanche, im- agerie sur la courbe de diffraction et topographie laminaire et ponctuelle, a été dévéloppée. Les désorientations angulaires, les largeurs à mi-hauteur et les intensités intégrées ont été mésurées dans les trois dimensions spatiales de l'échantillon et avec une résolution de 50× 50 × 50µm3. Les algorithmes d'analyse de données ont été écrits pour extraire des données des résultats quantitatifs, et pour calculer les neuf composantes du tenseur de courbure ainsi que la distortion entière des plans cristallins. Les résultats ont permis d'observer les premières étappes de la déformation de la glace. Par example la polygonisation d'un grain à été observée. / The viscoplastic deformation of ice is strongly anisotropic. The preferred glide system is on the basal plane. In a polycrystal each grain exhibits its own deformation direction. As a result the deformation of polycrystalline ice is associated with strain in- compatibilities, especially at the grain boundaries and the triple junction. The deforma- tion process was experimentally investigated by measuring crystal lattice distortions of single- and polycrystalline, artificially grown ice crystals. The experiments were benefic- ing from a synchrotron X-ray beam. A new experimental method, based on Bragg diffrac- tion imaging (X-ray topography) methods, as white beam X-ray diffraction topography, rocking curve imaging, section- and pinhole X-ray topography was used. Angular mis- orientations, full-width-half-maxima and integrated Bragg diffracted intensities have been measured along the three spatial dimensions of the sample and with a spatial resolution of around 50µm × 50µm × 50µm. Data analysis algorithms were written in order to extract quantitative results from the data and to calculate all nine components of the curvature ten- sor, as well as the entire lattice distortion in the sample. The results give an insight into the early stages of plastic deformation of ice, i.e. the polygonisation of a grain was observed.

Almost contact metric submersions constitute a class of Riemannian submersions whose total space is an almost contact metric manifold. Regarding the base space, two types are studied. Submersions of type I are those whose base space is an almost contact metric manifold while, when the base space is an almost Hermitian manifold, then the submersion is said to be of type II. After recalling the known notions and fundamental properties to be used in the sequel, relationships between the structure of the fibres with that of the total space are established. When the fibres are almost Hermitian manifolds, which occur in the case of a type I submersions, we determine the classes of submersions whose fibres are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal (almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of submersions of type I based upon the structure of the fibres. Concerning the fibres of a type II submersions, which are almost contact metric manifolds, we discuss how they inherit the structure of the total space. Considering the curvature property on the total space, we determine its corresponding on the fibres in the case of a type I submersions. For instance, the cosymplectic curvature property on the total space corresponds to the Kähler identity on the fibres. Similar results are obtained for Sasakian and Kenmotsu curvature properties. After producing the classes of submersions with minimal, superminimal or umbilical fibres, their impacts on the total or the base space are established. The minimality of the fibres facilitates the transference of the structure from the total to the base space. Similarly, the superminimality of the fibres facilitates the transference of the structure from the base to the total space. Also, it is shown to be a way to study the integrability of the horizontal distribution. Totally contact umbilicity of the fibres leads to the asymptotic directions on the total space. Submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)

Made available in DSpace on 2014-07-29T16:02:18Z (GMT). No. of bitstreams: 1 Dissertacao Fernando Soares de Carvalho.pdf: 3468325 bytes, checksum: 30df6cf936483cf5aec035b1bdd9d208 (MD5) Previous issue date: 2011-01-28 / NOTE: Because some programs do not copy symbols, formulas, etc... to view the summary and the contents of the file, click on PDF - dissertation on the bottom of the screen. / OBS: Como programas não copiam certos símbolos, fórmulas... etc, para visualizar o resumo e o todo o arquivo, click em PDF - dissertação na parte de baixo da tela.

Almost contact metric submersions constitute a class of Riemannian submersions whose total space is an almost contact metric manifold. Regarding the base space, two types are studied. Submersions of type I are those whose base space is an almost contact metric manifold while, when the base space is an almost Hermitian manifold, then the submersion is said to be of type II. After recalling the known notions and fundamental properties to be used in the sequel, relationships between the structure of the fibres with that of the total space are established. When the fibres are almost Hermitian manifolds, which occur in the case of a type I submersions, we determine the classes of submersions whose fibres are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal (almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of submersions of type I based upon the structure of the fibres. Concerning the fibres of a type II submersions, which are almost contact metric manifolds, we discuss how they inherit the structure of the total space. Considering the curvature property on the total space, we determine its corresponding on the fibres in the case of a type I submersions. For instance, the cosymplectic curvature property on the total space corresponds to the Kähler identity on the fibres. Similar results are obtained for Sasakian and Kenmotsu curvature properties. After producing the classes of submersions with minimal, superminimal or umbilical fibres, their impacts on the total or the base space are established. The minimality of the fibres facilitates the transference of the structure from the total to the base space. Similarly, the superminimality of the fibres facilitates the transference of the structure from the base to the total space. Also, it is shown to be a way to study the integrability of the horizontal distribution. Totally contact umbilicity of the fibres leads to the asymptotic directions on the total space. Submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)

Nous nous intéressons à la conception des mains mécaniques anthropomorphes destinées à manipuler des objets dans un environnement humain. Via l'analyse du mouvement de sujets humains lors d'une tâche de manipulation de référence, nous proposons une méthode pour évaluer la capacité des mains robotiques à manipuler les objets. Nous montrons comment les rapports de couplage angulaires entre les articulations et les limites articulaires, influent sur l'aptitude à manipuler dynamiquement des objets. Nous montrons également l'impact du poignet sur les tâches de manipulation rapides. Nous proposons une stratégie pour calculer les forces de manipulation en bout de doigts et dimensionner les moteurs d'un tel préhenseur. La méthode proposée est dépendante de la tâche visée et s'adapte à tout type de mouvement dès lors qu'il peut être capturé et analysé. Dans une deuxième partie, consacrée aux robots manipulateurs, nous élaborons des algorithmes de commande optimale. En considérant l'énergie cinétique du robot comme une métrique, le modèle dynamique est formulé sous forme tensorielle dans le cadre de la géométrie Riemannienne. La discrétisation temporelle est basée sur les Éléments Finis d'Hermite. Nous intégrons les équations de Lagrange du mouvement par une méthode de perturbation. Des exemples de simulation illustrent la superconvergence de la technique d'Hermite. Le critère de contrôle est choisi indépendant des paramètres de configuration. Les équations de la commande associées aux équations du mouvement se révèlent covariantes. La méthode de commande optimale proposée consiste à minimiser la fonction objective correspondant au critère invariant sélectionné. / We focus on the design of anthropomorphous mechanical hands destined to manipulate objects in a human environment. Via the motion analysis of a reference manipulation task performed by human subjects, we propose a method to evaluate a robotic hand manipulation capacities. We demonstrate how the angular coupling between the fingers joints and the angular limits affect the hands potential to manipulate objects. We also show the influence of the wrist motions on the manipulation task. We propose a strategy to calculate the fingertip manipulation forces and dimension the fingers motors. In a second part devoted to articulated robots, we elaborate optimal control algorithms. Regarding the kinetic energy of the robot as a metric, the dynamic model is formulated tensorially in the framework of Riemannian geometry. The time discretization is based on the Hermite Finite Elements.A time integration algorithm is designed by implementing a perturbation method of the Lagrange's motion equations. Simulation examples illustrate the superconvergence of the Hermite's technique. The control criterion is selected to be coordinate free. The control equations associated with the motion equations reveal to be covariant. The suggested control method consists in minimizing the objective function corresponding to the selected invariant criterion.