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ASPECTS OF THE GEOMETRY OF METRICAL CONNECTIONSWells, Matthew J. 01 January 2009 (has links)
Differential geometry is about space (a manifold) and a geometric structure on that space. In Riemann’s lecture (see [17]), he stated that “Thus arises the problem, to discover the matters of fact from which the measure-relations of space may be determined...”. It is key then to understand how manifolds differ from one another geometrically. The results of this dissertation concern how the geometry of a manifold changes when we alter metrical connections. We investigate how diverse geodesics are in different metrical connections. From this, we investigate a new class of metrical connections which are dependent on the class of smooth functions. Specifically, we fix a Riemannian metric and investigate the geometry of the manifold when we change the metrical connections associated with the fixed Riemannian metric. We measure the change in the Riemannian curvatures associated with this new class of metrical connections, and then give uniqueness and existence criterion for curvature of compact 2-manifolds. These results depend on the use of Hodge Theory and ultimately on the function f we choose to define a metrical connection.
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Le groupe conforme des structures pseudo-riemanniennes / The conformal group of pseudo-Riemannian structuresPecastaing, Vincent 12 December 2014 (has links)
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.
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Neural Networks and the Natural GradientBastian, Michael R. 01 May 2010 (has links)
Neural network training algorithms have always suffered from the problem of local minima. The advent of natural gradient algorithms promised to overcome this shortcoming by finding better local minima. However, they require additional training parameters and computational overhead. By using a new formulation for the natural gradient, an algorithm is described that uses less memory and processing time than previous algorithms with comparable performance.
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Analysis on a Class of Carnot Groups of Heisenberg TypeMcNamee, Meagan 14 July 2005 (has links)
In this thesis, we examine key geometric properties of a class of Carnot groups of Heisenberg type. After first computing the geodesics, we consider some partial differential equations in such groups and discuss viscosity solutions to these equations.
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Statistical Computing on Manifolds for Computational AnatomyPennec, Xavier 18 December 2006 (has links) (PDF)
During the last decade, my main research topic was on medical image analysis, and more particularly on image registration. However, I was also following in background a more theoretical research track on statistical computing on manifolds. With the recent emergence of computational anatomy, this topic gained a lot of importance in the medical image analysis community. During the writing of this habilitation manuscript, I felt that it was time to present a more simple and uni ed view of how it works and why it can be important. This is why the usual short synthesis of the habilitation became a hundred pages long text where I tried to synthesizes the main notions of statistical computing on manifolds with application in registration and computational anatomy. Of course, this synthesis is centered on and illustrated by my personal research work.
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Διαρμονικές υποπολλαπλότητες της σφαίρας S3 / Biharmonic submanifolds of sphere S3Σερεμετάκη, Στέλλα 30 August 2007 (has links)
Αντικείμενο της εργασίας αυτής είναι η αναζήτηση των διαρμονικών υποπολλαπλοτήτων της σφαίρας S3. Η μέθοδος που εφαρμόζεται συνδέεται με την αρχή του λογισμού των μεταβολών.
Γίνεται σύντομη ανάλυση της μεθοδολογίας του λογισμού των μεταβολών και εφαρμογή αυτής σε γνωστές θεωρίες μεταξύ των οποίων είναι οι αρμονικές και διαρμονικές απεικονίσεις. Ορίζουμε τις έννοιες των αρμονικών και διαρμονικών απεικονίσεων μεταξύ δύο πολλαπλοτήτων Riemann και δίνουμαι παραδείγματα τέτοιων απεικονίσεων. Τέλος, προσδιορίζουμαι τις διαρμονικές καμπύλες και τις διαρμονικές επιφάνειες της σφαίρας S3.
Οι κεντρικές μας αναφορές είναι οι εργασίες :
(1) Biharmonic submanifolds in spheres, Israel.J.Math.,130(2002), 109-123, των R.Caddeo, S. Montaldo και C .Oniciuic.
(2) A report on harmonic maps, Bull. London Math. Soc. 10(1978), 1-68 των J. Eells και L.Lemaire. / The object of this project is the investigation of the biharmonic submanifolds of sphere S3. The method we apply is the variational method. We shortly analyse the method of variations and we describe some theorys as they derived by this method.
Between those theorys are the harmonic and biharmonic maps. We define the notions of harmonic and biharmonic maps between two Riemannian manifolds and we introduce some examples. Finally, we allocate the biharmonic curves and surfaces of sphere S3. The central references are:
(1) Biharmonic submanifolds in spheres, Israel.J.Math.,130(2002), 109-123, των R.Caddeo, S. Montaldo και C .Oniciuic.
(2) A report on harmonic maps, Bull. London Math. Soc. 10(1978), 1-68 των J. Eells και L.Lemaire.
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Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric MorphometricsGao, Tingran January 2015 (has links)
<p>We introduce Hypoelliptic Diffusion Maps (HDM), a novel semi-supervised machine learning framework for the analysis of collections of anatomical surfaces. Triangular meshes obtained from discretizing these surfaces are high-dimensional, noisy, and unorganized, which makes it difficult to consistently extract robust geometric features for the whole collection. Traditionally, biologists put equal numbers of ``landmarks'' on each mesh, and study the ``shape space'' with this fixed number of landmarks to understand patterns of shape variation in the collection of surfaces; we propose here a correspondence-based, landmark-free approach that automates this process while maintaining morphological interpretability. Our methodology avoids explicit feature extraction and is thus related to the kernel methods, but the equivalent notion of ``kernel function'' takes value in pairwise correspondences between triangular meshes in the collection. Under the assumption that the data set is sampled from a fibre bundle, we show that the new graph Laplacian defined in the HDM framework is the discrete counterpart of a class of hypoelliptic partial differential operators.</p><p>This thesis is organized as follows: Chapter 1 is the introduction; Chapter 2 describes the correspondences between anatomical surfaces used in this research; Chapter 3 and 4 discuss the HDM framework in detail; Chapter 5 illustrates some interesting applications of this framework in geometric morphometrics.</p> / Dissertation
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Aplicações harmonicas, holomorfas e metricas(1,2)-simpleticas em variedades bandeira / Harmonic maps, holomorphic maps and (1-2)-sympletic metrics on flag manifoldsBressan, João Paulo, 1983- 03 June 2007 (has links)
Orientador: Caio Jose Colleti Negreiros / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T09:41:20Z (GMT). No. of bitstreams: 1
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Previous issue date: 2007 / Resumo: O objetivo deste trabalho é estudar a relação existente entre holomorfia e harmonicidade de aplicações f : M 2 (IF; J; ds2? ), onde M 2 é uma superfície de Riemann compacta, orientável e IF é a variedade bandeira maximal. Para isto, apresentamos parte da teoria geral de aplicações harmônicas e holomorfas, necessária para demonstrar o teorema de Lichnerowicz. Uma de suas conseqüências é uma ferramenta importante neste estudo, pois fornece o seguinte critério: se f é J-holomorfa e ds2? é (1,2)-simplética, então f é harmônica. Portanto, também estamos interessados em descrever as métricas (1,2)-simpléticas nas variedades bandeira. Primeiramente, no caso geométrico, estudamos a variedade bandeira complexa maximal de subespaços encaixados IF(n). Posteriormente, este estudo é generalizado para outras variedades bandeiras maximais IF, definidas através de álgebras de Lie semi-simples complexas. Ainda, demonstramos o teorema de Burstall-Salamon, que fornece propriedades da estrutura quase complexa invariante J através de um torneio ?J associado. Finalmente, exibimos as equações de Cauchy-Riemann e de Euler-Lagrange para estas aplicações, e apresentamos exemplos de famílias de funções equi-harmônicas / Abstract: The goal of this work is to study the relationship bettwen holomorphicity and harmonicity of maps f: M 2 (IF; J; ds2? ), where M 2 is a compact, orientable Riemann surface and IF is the full-flag manifold. With this pourpose, we present part of the general holomorphic/harmonic maps theory, which is necessary to prove the Lichnerowicz theorem.It states like consequence a criterion, which is an important tool in this study: if f is J-holomorphic and ds2? é (1,2)-symplectic, then f is harmonic. Therefore, we are interested in to describe (1,2)-symplectc metrics on the flag manifold.Firstly, in the geometrical case, we study the complex full-flag manifold IF(n). Later, we generalize this study to other full-flag manifolds IF, which is defined through complex semisimple Lie algebras. Also, we prove the Burstall-Salamon theorem, which gives some properties of the almost complex structure J through an associated tournament ?J. Finally, we show-up the Cauchy-Riemann equations and the Euler-Lagrange equations to this maps, and present examples of families of equi-harmonic maps / Mestrado / Mestre em Matemática
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Auto-valores do operador de Dirac e do laplaciano de Dobeault / Eigenvalues of Dirac operator and Dolbeault laplacianLeão, Rafael de Freitas, 1979- 19 April 2007 (has links)
Orientador: Marcos Benevenuto Jardim / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T16:20:15Z (GMT). No. of bitstreams: 1
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Previous issue date: 2007 / Resumo: Nesta tese estudamos basicamente como o acoplamento por uma conexão arbitraria influencia o comportamento do espectro do operador de Dirac, real e complexo. Atraves dos resultados classicos da literatura, e destes resultados vemos que, de modo geral, estruturas geometricas influenciam o espectro do operador de Dirac, acoplado ou não. Embora exista uma grande literatura a respeito de estruturas geometricas e o operador de Dirac, sobretudo para o operador não acoplado, existem alguns casos, possivelmente bastante interessantes, que não foram considerados. Com o recente desenvolvimento de geometria complexa generalizada, podemos nos perguntar sobre a possibilidade de definirmos operadores de Dirac neste contexto e se isto traz resultados novos ou entendimento sobre resultados ja conhecidos. Por se tratar de uma area recente existem varios problemas envolvidos na tentativa de estudarmos operadores de Dirac sobre variedades com estruturas complexas generalizadas. O proprio conceito de conexão para este tipo de geometria ainda não e muito claro, uma vez que não assumimos a priori uma metrica na variedade base não podemos considerar a conexão Levi-Civita, ficando a pergunta que se neste contexto existe alguma conexão natural analoga a conexão de Levi-Civita. Outra questão importante e com relação ao fibrado de spinores. No caso de variedades riemannianas a maneira mais usual de construirmos fibrados de Dirac e atraves de uma estrutura Spin na variedade base. Porem este tipo de estrutura tambem e definida em termos de uma metrica ficando a pergunta de como poderíamos construir fibrados de Dirac de maneira natural sobre uma variedade complexa generalizada. Caso seja possível respondermos estas questões podemos falar em operadores de Dirac sobre variedades complexas generalizadas. Podendo, a partir dai, investigar formulas do tipo Weitzenbock e o comportamento do espectro do operador de Dirac. Alem disso podemos nos perguntar se este tipo de operador e de fato um objeto totalmente novo ou se o mesmo se relaciona com operadores conhecidos da variedade base. Outro situação pouco explorada na literatura e a de operadores de Dirac sobre variedades algebricas imersas em CPn. Na literatura existem artigos, [5, 16], que exploram sobretudo estruturas Spin e spinores. Mas não existe tentativas de usar explicitamente que certas variedades podem ser consideradas como variedades algebricas imersas em CPn para tentar obter estimativas mais finas para o espectro do operador de Dirac, como por exemplo e feito para subvariedades Lagrangianas em [8]. Para considerarmos este problema devemos entender como considerar explicitamente que estamos lidando com variedades algebricas imersas em CPn. É possível que existam duas formas de fazermos isto. A primeira e aparentemente mais direta e considerar a imersão em si, na linha do que foi feito com subvariedades Lagrangianas em [8], e estudar propriedades da mesma. Para fazermos isto é possiível que tenhamos que restringir a classe de variedades em questão. A segunda forma, que parece ser um pouco mais delicada, é tentar escrever o operador de Dirac de forma a levar em consideração a estrutura algebrica da variedade. Pode ser possível que escrevendo o operador de Dirac na linguagem algebrica obtenhamos informações que nos permitirão encontrar estimativas para o espectro do mesmo / Abstract: Not informed. / Doutorado / Geometria / Doutor em Matemática
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Geometria diferencial em grupos de Lie / Differential geometry on Lie groupsCorrea, Eder de Moraes, 1986- 05 February 2013 (has links)
Orientadores: Luiz Antonio Barrera San Martin, Lino Anderson da Silva Grama / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T20:52:19Z (GMT). No. of bitstreams: 1
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Previous issue date: 2013 / Resumo: Neste trabalho estudamos os aspectos geométricos dos grupos de Lie do ponto de vista da geometria Riemanniana, geometria Hermitiana e geometria Kähler, através das estruturas geométricas invariantes associadas. Exploramos resultados relacionados às curvaturas da variedade Riemanniana subjacente a um grupo de Lie através do estudo de sua álgebra de Lie correspondente. No contexto da geometria Hermitiana e geometria Kähler, para um caso concreto de grupo de Lie complexo, investigaram suas curvaturas seccionais holomorfas e verificamos a existência de uma estrutura pseudo-Kähler invariante por sua forma real compacta / Abstract: In this dissertation, we study the geometric aspects of Lie groups from the viewpoint of Riemannian geometry, Hermitian geometry, and Kähler geometry through its associated invariant geometric structures. We explore results related to curvatures of Riemannian manifold underlying a Lie group by studying its corresponding Lie algebra. In the context of Hermitian geometry and Kähler geometry, for a complex Lie group case, we investigate its holomorphic sectional curvatures and verify the existence of pseudo-Kähler structure invariant for its compact real form / Mestrado / Matematica / Mestre em Matemática
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