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Estruturas de indexação para domínios métricos são úteis para agilizar consultas por similaridade sobre dados complexos, tais como imagens, onde o custo computacional da comparação de dois itens de dados geralmente é alto. O estado da arte para executar consultas por similaridade está centrado na utilização dos chamados \"Métodos de Acesso Métrico\" (MAM). Tais métodos consideram os dados como elementos de um espaço métrico, onde apenas valem as propriedades fundamentais para que um espaço seja considerado métrico, onde a única informação que os MAMs utilizam é a medida de similaridade entre pares de elementos do domínio. No campo teórico, espaços métricos são extensamente estudados e servem de base para diversas áreas da Matemática. No entanto, a maioria dos trabalhos que têm sido desenvolvidos em Computação se restringem a utilizar as definições básicas desses espaços, e não foram encontrados estudos que explorem em mais profundidade os muitos conceitos teóricos existentes. Assim, este trabalho aplica conceitos teóricos importantes da Teoria de Espaços Métricos para desenvolver técnicas que auxiliem o tratamento e a manipulação dos diversos dados complexos, visando principalmente o desenvolvimento de métodos de indexação mais eficientes. É desenvolvida uma técnica para realizar um mapeamento de espaços métricos que leva à atenuação do efeito da maldição da dimensionalidade, a partir de uma aplicação lipschitziana real baseada em uma função de deformação do espaço das distâncias entre os elementos do conjunto. Foi mostrado que uma função do tipo exponecial deforma as distâncias de modo a diminuir os efeitos da maldição da dimensionalidade, melhorando assim o desempenho nas consultas. Uma segunda contribuição é o desenvolvimento de uma técnica para a imersão de espaços métricos, realizada de maneira a preservar a ordem das distâncias, possibilitando a utilização de propriedades no espaço de imersão. A imersão de espaços métricos no \' R POT. n\' possibilita a utilização da lei dos cossenos e assim viabiliza o cálculo de distâncias entre elementos e um hiperplano métrico, permitindo aumentar a agilidade à consultas por similaridade. O uso do hiperplano métrico foi exemplificado construindo uma árvore binária métrica, e também foi aplicado em um método de acesso métrico, a família MMH de métodos de acesso métrico, melhorando o particionamento do espaço dos dados / The access methods designed for metric domains are useful to answer similarity queries on any type of data, being specially useful to index complex data, such as images, where the computacional cost of comparison are high. The main mecanism used up to now to perform similarity queries is centered on \"Metric Acess Methods\" (MAM). Such methods consider data as elements that belong to a metric space, where only hold the properties that define the metric space. Therefore, the only information that a MAM can use is the similarity measure between pairs of elements in the domain. Metric spaces are extremelly well studied and is the basis for many mathematics areas. However, most researches from computer science are restrained to use the basic properties of metric spaces, not exploring the various existing theorical concepts. This work apply theoretical concepts of metric spaces to develop techniques aiding the treatment and manipulation of diverse complex data, aiming at developing more efficient indexing methods. A technique of mapping spaces was developed in order to ease the dimensionality curse effects, basing on a real lipschitz application that uses a stretching function that changes the distance space of elements. It was shown that an exponential function changes the distances space reducing the dimensionality curse effects, improving query operations. A second contribution is the developing of a technique based on metric space immersion, preserving the distances order between pairs of elements, allowing the usage of immersion space properties. The immersion of metric spaces into \'R POT. n\' allow the usage of the cossine law leading to the determination of distances between elements and a hiperplane, forming metric hiperplanes. The use of the metric hiperplanes lead to an improvement of query operations performance. The metric hiperplane itself formed the binary metric tree, and when applied to a metric access method, lead the formation of a family of metric access methods that improves the metric space particioning achieving faster similarity queries

Estruturas de indexação para domínios métricos são úteis para agilizar consultas por similaridade sobre dados complexos, tais como imagens, onde o custo computacional da comparação de dois itens de dados geralmente é alto. O estado da arte para executar consultas por similaridade está centrado na utilização dos chamados \"Métodos de Acesso Métrico\" (MAM). Tais métodos consideram os dados como elementos de um espaço métrico, onde apenas valem as propriedades fundamentais para que um espaço seja considerado métrico, onde a única informação que os MAMs utilizam é a medida de similaridade entre pares de elementos do domínio. No campo teórico, espaços métricos são extensamente estudados e servem de base para diversas áreas da Matemática. No entanto, a maioria dos trabalhos que têm sido desenvolvidos em Computação se restringem a utilizar as definições básicas desses espaços, e não foram encontrados estudos que explorem em mais profundidade os muitos conceitos teóricos existentes. Assim, este trabalho aplica conceitos teóricos importantes da Teoria de Espaços Métricos para desenvolver técnicas que auxiliem o tratamento e a manipulação dos diversos dados complexos, visando principalmente o desenvolvimento de métodos de indexação mais eficientes. É desenvolvida uma técnica para realizar um mapeamento de espaços métricos que leva à atenuação do efeito da maldição da dimensionalidade, a partir de uma aplicação lipschitziana real baseada em uma função de deformação do espaço das distâncias entre os elementos do conjunto. Foi mostrado que uma função do tipo exponecial deforma as distâncias de modo a diminuir os efeitos da maldição da dimensionalidade, melhorando assim o desempenho nas consultas. Uma segunda contribuição é o desenvolvimento de uma técnica para a imersão de espaços métricos, realizada de maneira a preservar a ordem das distâncias, possibilitando a utilização de propriedades no espaço de imersão. A imersão de espaços métricos no \' R POT. n\' possibilita a utilização da lei dos cossenos e assim viabiliza o cálculo de distâncias entre elementos e um hiperplano métrico, permitindo aumentar a agilidade à consultas por similaridade. O uso do hiperplano métrico foi exemplificado construindo uma árvore binária métrica, e também foi aplicado em um método de acesso métrico, a família MMH de métodos de acesso métrico, melhorando o particionamento do espaço dos dados / The access methods designed for metric domains are useful to answer similarity queries on any type of data, being specially useful to index complex data, such as images, where the computacional cost of comparison are high. The main mecanism used up to now to perform similarity queries is centered on \"Metric Acess Methods\" (MAM). Such methods consider data as elements that belong to a metric space, where only hold the properties that define the metric space. Therefore, the only information that a MAM can use is the similarity measure between pairs of elements in the domain. Metric spaces are extremelly well studied and is the basis for many mathematics areas. However, most researches from computer science are restrained to use the basic properties of metric spaces, not exploring the various existing theorical concepts. This work apply theoretical concepts of metric spaces to develop techniques aiding the treatment and manipulation of diverse complex data, aiming at developing more efficient indexing methods. A technique of mapping spaces was developed in order to ease the dimensionality curse effects, basing on a real lipschitz application that uses a stretching function that changes the distance space of elements. It was shown that an exponential function changes the distances space reducing the dimensionality curse effects, improving query operations. A second contribution is the developing of a technique based on metric space immersion, preserving the distances order between pairs of elements, allowing the usage of immersion space properties. The immersion of metric spaces into \'R POT. n\' allow the usage of the cossine law leading to the determination of distances between elements and a hiperplane, forming metric hiperplanes. The use of the metric hiperplanes lead to an improvement of query operations performance. The metric hiperplane itself formed the binary metric tree, and when applied to a metric access method, lead the formation of a family of metric access methods that improves the metric space particioning achieving faster similarity queries

We consider curves γ : [0, 1] → R3 endowed with an adapted orthonormal frame r : [0, 1] → SO(3). We wish to deform such framed curves (γ ,r) while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements ofr r T ), and a global constraint prescribing the initial and terminal values of γ and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve γ is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on (γ ,r) that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.

This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on Legendrian contact homology. In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So to equip the trees with orientations corresponds to orienting the determinant line bundle of the dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We define an  orientation of this line bundle and prove that it is well-defined in the limit. We also prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing the orientation of the trees, and we give an explicit description of this algorithm. In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to prove invariance of Legendrian contact homology for Legendrian knots over the integers can be derived analytically.  This is proved using the orientation scheme from Paper I together with a count of abstractly perturbed flow trees  of the Lagrangian cobordisms. In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in the standard symplectic plane.

Ce travail est relatif à la théorie des immersions et utilise des méthodes issues de la géométrie spinorielle, para-complexe et para-quaternionique. Les deux premières parties sont consacrées aux immersions conformes de surfaces pseudo-Riemanniennes. D'une part, nous étudions ce type d'immersions dans l'espace pseudo-Euclidien de dimension trois. Avec des méthodes de géométrie para-complexe et des représentations spinorielles réelles, l'équivalence entre les données d'une immersion conforme d'une surface de Lorentz dans $\mathbb{R}^{2,1}$ et de spineurs satisfaisant une équation de type Dirac est prouvée. D'autre part nous considérons des surfaces de Lorentz dans la pseudo-sphère $\mathbb{S}^{2,2}$: une bijection entre ces immersions et des sous-fibrés en droite para-quaternioniques du fibré $M\times\mathbb{H}^2$ est établie. Considérant une structure (para-)complexe particulière de ce fibré, la congruence pseudo-sphérique, et les champs de Hopf para-quaternioniques, nous définissons la fonctionnelle de Willmore de la surface et exprimons son énergie comme la somme de cette fonctionnelle et d'un invariant topologique. La dernière partie, plus générale, traite des fibrés vectoriels et immersions affines para-complexes. Nous introduisons la notion de fibré vectoriel para-holomorphe, et les sous-fibrés para-holomorphes et de type $(1,1)$ en termes de connections associées induites et de secondes formes fondamentales. Les équations fondamentales pour des décompositions générales de fibrés vectoriels munis d'une connexion sont étudiées dans le cas où certains des fibrés sont para-holomorphes afin d'obtenir des théorèmes d'existence et d'unicité pour des immersions affines para-complexes.

In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L^p continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H^r+1 (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H^r, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.

Made available in DSpace on 2015-05-15T11:46:02Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 666060 bytes, checksum: 1ad661f6cc42df5f3ee67db9a939af86 (MD5) Previous issue date: 2011-07-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we obtain Weierstrass-type representations for immersed surfaces in Heisenberg space, endowed with a left-invariant metric. We will consider the Riemannian and Lorentzian case. We will define two complex functions (spinors) satisfying a linear Dirac-type equation, obtaining thus a representation for immersed surfaces with prescribed mean curvature. The same will enable us write a representation of minimal immersion in terms of a harmonic Gauss map. / Neste trabalho obtemos uma representações tipo Weierstrass para superfícies imersas no espaço de Heisenberg, dotado com uma métrica invariante à esquerda. Consideraremos os casos Riemanniano e Lorentziano. Definimos duas funções complexas (spinors), satisfazendo uma equação linear tipo Dirac que usamos para obter uma representação para superfícies imersas com curvatura média prescrita. A mesma possibilita escrever uma representação de imersões mínimas em termos de uma aplicação de Gauss harmônica.

Dans cette thèse, nous étudions le spectre conforme d'une surface fermée et le spectre de Steklov conforme d'une surface compacte à bord et leur application à la géométrie conforme et à la topologie. Soit (Σ, c) une surface fermée munie d'une classe conforme c. Alors la k-ième valeur propre conforme est définie comme Λ_k(Σ,c)=sup{λ_k(g) Aire(Σ,g)| g ∈ c), où λ_k(g) est la k-ième valeur propre de l'operateur de Laplace-Beltrami de la métrique g sur Σ. Notons que nous commeçons par λ_0(g) = 0. En prennant le supremum sur toutes les classes conformes C sur Σ on obtient l'invariant topologique suivant de Σ: Λ_k(Σ)=sup{Λ_k(Σ,c)| c ∈ C}. D'après l'article [65], les quantités Λ_k(Σ, c) et Λ_k(Σ) sont bien définies. Si une métrique g sur Σ satisfait λ_k(g) Aire(Σ, g) = Λ_k(Σ), alors on dit que g est maximale pour la fonctionnelle λ_k(g) Aire(Σ, g). Dans l'article [73], il a été montré que les métriques maximales pour λ_1(g) Aire(Σ, g) peuvent au pire avoir des singularités coniques. Dans cette thèse nous montrons que les métriques maximales pour les fonctionnelles λ_1(g) Aire(T^2, g) et λ_1(g) Aire(KL, g), où T^2 et KL dénotent le 2-tore et la bouteille de Klein, ne peuvent pas avoir de singularités coniques. Ce résultat découle d'un théorème de classification de classes conformes par des métriques induites d'une immersion minimale ramifiée dans une sphère ronde aussi montré dans cette thèse. Un autre invariant que nous étudions dans cette thèse est le k-ième invariant de Friedlander-Nadirashvili défini comme: I_k(Σ) = inf{Λ_k(Σ, c)| c ∈ C}. L'invariant I_1(Σ) a été introduit dans l'article [34]. Dans cette thèse nous montrons que pour toute surface orientable et pour toute surface non-orientable de genre impaire I_k(Σ)=I_k(S^2) et pour toute surface non-orientable de genre paire I_k(RP^2) ≥ I_k(Σ)>I_k(S^2). Ici S^2 et RP^2 dénotent la 2-sphère et le plan projectif. Nous conjecturons que I_k(Σ) sont des invariants des cobordismes des surfaces fermées. Le spectre de Steklov conforme est défini de manière similaire. Soit (Σ, c) une surface compacte à bord non vide ∂Σ, alors les k-ièmes valeurs propres de Steklov conformes sont définies comme: σ*_k(Σ, c)=sup{σ_k(g) Longueur(∂Σ, g)| g ∈ c}, où σ_k(g) est la k-ième valeur propre de Steklov de la métrique g sur Σ. Ici nous supposons que σ_0(g) = 0. De façon similaire au problème fermé, on peut définir les quantités suivantes: σ*_k(Σ)=sup{σ*_k(Σ, c)| c ∈ C} et I^σ_k(Σ)=inf{σ*_k(Σ, c)| c ∈ C}. Les résultats de l'article [16] impliquent que toutes ces quantités sont bien définies. Dans cette thèse on obtient une formule pour la limite de σ*_k(Σ, c_n) lorsque la suite des classes conformes c_n dégénère. Cette formule implique que pour toute surface à bord I^σ_k(Σ)= I^σ_k(D^2), où D^2 dénote le 2-disque. On remarque aussi que les quantités I^σ_k(Σ) sont des invariants des cobordismes de surfaces à bord. De plus, on obtient une borne supérieure pour la fonctionnelle σ^k(g) Longueur(∂Σ, g), où Σ est non-orientable, en terme de son genre et le nombre de composants de bord. / In this thesis, we study the conformal spectrum of a closed surface and the conformal Steklov spectrum of a compact surface with boundary and their application to conformal geometry and topology. Let (Σ,c) be a closed surface endowed with a conformal class c then the k-th conformal eigenvalue is defined as Λ_k(Σ,c)=sup{λ_k(g) Aire(Σ,g)| g ∈ c), where λ_k(g) is the k-th Laplace-Beltrami eigenvalue of the metric g on Σ. Note that we start with λ_0(g) = 0 Taking the supremum over all conformal classes C on Σ one gets the following topological invariant of Σ: Λ_k(Σ)=sup{Λ_k(Σ,c)| c ∈ C}. It follows from the paper [65] that the quantities Λ_k(Σ, c) and Λ_k(Σ) are well-defined. Suppose that for a metric g on Σ the following identity holds λ_k(g) Aire(Σ, g) = Λ_k(Σ). Then one says that the metric g is maximal for the functional λ_k(g) Aire(Σ, g). In the paper [73] it was shown that the maximal metrics for the functional λ_1(g) Aire(Σ, g) at worst can have conical singularities. In this thesis we show that the maximal metrics for the functionals λ_1(g) Aire(T^2, g) and λ_1(g) Aire(KL, g), where T^2 and KL stand for the 2-torus and the Klein bottle respectively, cannot have conical singularities. This result is a corollary of a conformal class classification theorem by metrics induced from a branched minimal immersion into a round sphere that we also prove in the thesis. Another invariant that we study in this thesis is the k-th Friedlander-Nadirashvili invariant defined as: I_k(Σ) = inf{Λ_k(Σ, c)| c ∈ C}. The invariant I_1(Σ) was introduced in the paper [34]. In this thesis we prove that for any orientable surface and any non-orientable surface of odd genus I_k(Σ)=I_k(S^2) and for any non-orientable surface of even genus I_k(RP^2) ≥ I_k(Σ)>I_k(S^2). Here S^2 and RP^2 denote the 2-sphere and the projective plane respectively. We also conjecture that I_k(Σ) are invariants of cobordisms of closed manifolds. The conformal Steklov spectrum is defined in a similar way. Let (Σ, c) be a compact surface with non-empty boundary ∂Σ then the k-th conformal Steklov eigenvalues is defined by the formula: σ*_k(Σ, c)=sup{σ_k(g) Longueur(∂Σ, g)| g ∈ c}, where σ_k(g) is the k-th Steklov eigenvalue of the metric g on Σ. Here we suppose that σ_0(g) = 0. Similarly to the closed problem one can define the following quantities: σ*_k(Σ)=sup{σ*_k(Σ, c)| c ∈ C} and I^σ_k(Σ)=inf{σ*_k(Σ, c)| c ∈ C}. The results of the paper [16] imply that all these quantities are well-defined. In this thesis we obtain a formula for the limit of the k-th conformal Steklov eigenvalue when the sequence of conformal classes degenerates. Using this formula we show that for any surface with boundary I^σ_k(Σ)= I^σ_k(D^2), where D^2 stands for the 2-disc. We also notice that I^σ_k(Σ) are invariants of cobordisms of surfaces with boundary. Moreover, we obtain an upper bound for the functional σ^k(g) Longueur(∂Σ, g), where Σ is non-orientable, in terms of its genus and the number of boundary components.

Un problema clásico en geometría lorentziana es la descripción de las inmersiones isométricas entre los espacios lorentzianos de curvatura constante. En este trabajo nos centramos en la clasificación de las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter tridimensional. Damos una fórmula de representación de estas inmersiones en términos de pares de curvas (con posibles singularidades) en el plano hiperbólico. Esto nos permite resolver los problemas propuestos por Dajczer y Nomizu en 1981. De entre todas las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter, algunas de ellas corresponden a toros lorentzianos (los ejemplos más sencillos son los toros de Hopf). Como aplicación de nuestra anterior descripción, probamos que todos estos toros pueden obtenerse a partir de dos curvas cerradas en el espacio hiperbólico. Finalmente, demostramos que los toros de Hopf son los únicos toros llanos lorentzianos inmersos en una amplia familia de sumersiones de Killing lorentzianas tridimensionales. / A classical problem in Lorentzian geometry is the description of the isometric immersions between Lorentzian spaces of constant curvature. We investigate the problem of classifying the isometric immersion from the Lorentz plane into the three-dimensional anti-de Sitter space, providing a representation formula of these isometric immersions in terms of pairs of curves (possibly with singularities) in the hyperbolic plane. We then give an answer to the open problems proposed by Dajczer and Nomizu in 1981. Among all isometric immersions of the Lorentz plane into the anti-de Sitter space, some of them are actually Lorentzian tori (the basic examples are the Hopf tori). As an application of our previous description, we prove that any such torus can be recovered from two closed curves in the hyperbolic plane. Finally, we prove that Lorentzian Hopf tori are the only immersed Lorentzian flat tori in a wide family of Lorentzian three-dimensional Killing submersions.