abstract: The data explosion in the past decade is in part due to the widespread use of rich sensors that measure various physical phenomenon -- gyroscopes that measure orientation in phones and fitness devices, the Microsoft Kinect which measures depth information, etc. A typical application requires inferring the underlying physical phenomenon from data, which is done using machine learning. A fundamental assumption in training models is that the data is Euclidean, i.e. the metric is the standard Euclidean distance governed by the L-2 norm. However in many cases this assumption is violated, when the data lies on non Euclidean spaces such as Riemannian manifolds. While the underlying geometry accounts for the non-linearity, accurate analysis of human activity also requires temporal information to be taken into account. Human movement has a natural interpretation as a trajectory on the underlying feature manifold, as it evolves smoothly in time. A commonly occurring theme in many emerging problems is the need to \emph{represent, compare, and manipulate} such trajectories in a manner that respects the geometric constraints. This dissertation is a comprehensive treatise on modeling Riemannian trajectories to understand and exploit their statistical and dynamical properties. Such properties allow us to formulate novel representations for Riemannian trajectories. For example, the physical constraints on human movement are rarely considered, which results in an unnecessarily large space of features, making search, classification and other applications more complicated. Exploiting statistical properties can help us understand the \emph{true} space of such trajectories. In applications such as stroke rehabilitation where there is a need to differentiate between very similar kinds of movement, dynamical properties can be much more effective. In this regard, we propose a generalization to the Lyapunov exponent to Riemannian manifolds and show its effectiveness for human activity analysis. The theory developed in this thesis naturally leads to several benefits in areas such as data mining, compression, dimensionality reduction, classification, and regression. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2016

We define hypersato structures: these structures admit three inequivalent Sasakian structures such that each of these structures shares a common Reeb vector field and a common contact form with the others two. It is interestingto notice that hypersato manifolds can be viewed as U(1) principal orbibundles with base space a 4n-dimensional hyperkahler orbifold. We also discuss some results on the moduli problem of these structures. / Definimos variedades hipersato: estas variedades admiten tres estructuras del tipo Sasaki inequivalentes de tal manera que estas tres estructuras poseen un campo vectorial del tipo Reeb y una forma de contacto en común. Variedades que admiten estructura hipersato pueden considerarse como espacios totales de un fibrado principal U(1) del tipo orbifold, donde el espacio base admite una métrica singular hiperkahler. Discutimos tambien algunos resultados acerca del espacio moduli de variedades admitiendo estas estructuras.

Orientador: Pedro Jose Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T12:34:53Z (GMT). No. of bitstreams: 1 Felix_LucianoVianna_M.pdf: 566321 bytes, checksum: f717034fada0c65f1b886ba7bd821902 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho estudamos a geometria dos caminhos em grupos de Lie usando a exponencial estocástica e o logaritmo estocástico. Apresentamos as construções geométricas do espaço tangente, uma métrica e uma conexão natural as caminhos em grupos de Lie. Finalmente apresentamos uma situação em que essa conexão é Levi-Civita e outra que não é / Abstract: In this work, we study the path geometry in Lie groups using the stochastic exponential and the stochastic logarithm. We show the geometric constructions of tangent space, one metric and one natural conection of Lie groups valued path. Finelly we show one situation that this conection is Levi-Civita and another one that is not / Mestrado / Geometria / Mestre em Matemática

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2014-10-23T20:04:48Z No. of bitstreams: 2 Dissertação - Hiuri Fellipe Santos dos Reis - 2013.pdf: 1601406 bytes, checksum: f2663891a9c0968329f2f913ada41d9e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2014-10-23T20:05:03Z (GMT) No. of bitstreams: 2 Dissertação - Hiuri Fellipe Santos dos Reis - 2013.pdf: 1601406 bytes, checksum: f2663891a9c0968329f2f913ada41d9e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-10-23T20:05:03Z (GMT). No. of bitstreams: 2 Dissertação - Hiuri Fellipe Santos dos Reis - 2013.pdf: 1601406 bytes, checksum: f2663891a9c0968329f2f913ada41d9e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-03-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we present a study on locally conformally flat gradient steady Ricci solitons which is based on a Huai Dong-Cao and Qing Chen’s article, where they was classified the n-dimensional (n ≥ 3) complete noncompact locally conformally flat gradient steady Ricci solitons. In particular, we prove that a complete noncompact non-flat locally conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton. / Neste trabalho apresentamos um estudo dos sólitons de Ricci gradiente steady localmente conformemente flat, baseado no trabalho de Huai-Dong Cao e Qiang Chen, onde são classificados os sólitons de Ricci gradiente steady n-dimensionais (n ≥ 3), completos, não-compactos e localmente conformemente flat. Em particular provamos que um sóliton de Ricci gradiente steady completo, não-compacto, não-flat e localmente conformemente flat é, a menos de homotetia, o sóliton de Bryant.

Apresentaremos alguns resultados de rigidez e de bifurcação para soluções do problema de Yamabe em variedades produto com bordo. / We will discuss some rigidity and bifurcation results for solutions of the Yamabe problem in product manifolds with boundary.

Orientador: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-04T02:49:45Z (GMT). No. of bitstreams: 1 SilvaJunior_RinaldoVieirada_M.pdf: 1586291 bytes, checksum: 8d01bdf434ecba2fb62a57725c46dd4a (MD5) Previous issue date: 2005 / Resumo: Nesta dissertação fizemos um estudo da teoria de difusão em variedades Finsler, onde abor-damos o transporte paralelo estocástico, desenvolvimento estocástico de Cartan e Movimento Browniano. O objetivo principal é obter uma descrição mais geométrica dos objetos citados acima ainda que por enquanto em coordenadas locais e assim termos um paralelo entre o cálculo estocástico em variedades Riemannianas e variedades Finsler / Abstract: In this work we study diffusion theory in Finsler manifolds. It includes the stochastic par-allel transport, stochastic Cartan development and Brownian motion. The main objective is to provide a geometric description of the objects mentioned and 50 to draw a compari-50n between stochastic calculus in Riemannian manifolds and stochastic calculus in Finsler manifolds / Mestrado / Matematica / Mestre em Matemática

Given a compact smooth manifold Mn without boundary and n ≥ 3, the Lp-norm of the curvature tensor, defines a Riemannian functional on the space of Riemannian metrics with unit volume M1. Consider C2,α-topology on M1 Rp remains invariant under the action of the group of diffeomorphisms D of M. So, Rp is defined on M1/ D. Our first result is that Rp restricted to the space M1/D has strict local minima at Riemannian metrics with constant sectional curvature for certain values of p. The product of spherical space forms and the product of compact hyperbolic manifolds are also critical point for Rp if they are product of same dimensional manifolds. We prove that these spaces are strict local minima for Rp restricted to M1/D. Compact locally symmetric isotropy irreducible metrics are critical points for Rp. We give a criteria for the local minima of Rp restricted to the conformal class of metrics of a given irreducible symmetric metric. We also prove that the metrics with constant bisectional curvature are strict local minima for Rp restricted to the space of Kahlar metrics with unite volume quotient by D. Next we consider the Riemannian functional given by In [GV], M. J. Gursky and J. A. Viaclovsky studied the local properties of the moduli space of critical metrics for the functional Ric2.We generalize their results for any p > 0.

In Riemannian geometry, the fundamental fact is that there exists a unique torsion-free connection (called the Levi-Civita connection) compatible with the Riemannian metric g, i.e. having the property ∇g = 0. In projective geometry, the class of covariant derivatives defining the geometry is fixed and all these covariant derivatives have the same class of (non- parametrized) geodesics. Old (and non-trivial) problem is to find whether these curves are geodesics of a (pseudo-)Riemannian metric. Such projective structures are called metrizable. Surprisingly enough, U. Dini and R. Liu- oville found in 19th century that the metrizability problem leads to a system of linear PDE's. In the last years, there were several papers dealing with these problems. The projective geometry is a representative example of the so called parabolic geometries (for full description, see the recent monograph by A. Čap and J. Slovák). It was realized recently that the corresponding linear metrizability operator is a special example of the so called first BGG operator. The flat model of projective geometry is the (real) projective space. In this more general context, the metrizability problem for (pseudo- )Riemannian geometries is naturally generalized to the sub-Riemannian situation. In the recent preprint, D.Calderbank, J....

Nous étudions dans cette thèse la génération de maillages anisotropes basée sur la triangulation de Delaunay et le diagramme de Voronoi. Nous considérons tout d'abord les maillages anisotropes localement uniformes, développés par Boissonnat, Wormser et Yvinec. Bien que l'aspect théorique de cette approche soit connu, son utilité pratique n'a été que peu explorée. Une étude empirique exhaustive est présentée et révèle les avantages, mais aussi les inconvénients majeurs de cette méthode. Dans un second temps, nous étudions les diagrammes de Voronoi anisotropes définis par Labelle et Shewchuk. Nous donnons des conditions suffisantes sur un ensemble de points pour que le dual du diagramme soit une triangulation plongée en toute dimension ; un algorithme générant de tels ensembles est conçu. Ce diagramme est utilisé pour concevoir un algorithme qui génère efficacement un maillage anisotrope pour des domaines de dimension intrinsèque faible plongés dans des espaces de dimension large. Notre algorithme est prouvable, mais les résultats sont décevants. Enfin, nous présentons le diagramme de Voronoi Riemannien discret, qui utilise des avancées récentes dans l'estimation de distances géodésiques et dont le calcul est grandement accéléré par l'utilisation d'un graphe anisotrope. Nous donnons des conditions suffisantes pour que notre structure soit combinatoirement équivalente au diagramme de Voronoi Riemannien et que son dual utilisant des simplexes droits mais aussi courbes est une triangulation plongée en toute dimension. Nous obtenons de bien meilleurs résultats que pour nos autres techniques, mais dont l'utilité reste limitée / In this thesis, we study the generation of anisotropic meshes using the concepts of Delaunay triangulations and Voronoi diagrams. We first consider the framework of locally uniform anisotropic meshes introduced by Boissonnat, Wormser and Yvinec. Despite known theoretical guarantees, the practicality of this approach has only been hardly studied. An exhaustive empirical study is presented and reveals the strengths but also the overall impracticality of the method. In a second part, we investigate the anisotropic Voronoi diagram introduced by Labelle and Shewchuk and give conditions on a set of seeds such that the corresponding diagram has a dual that is an embedded triangulation in any dimension; an algorithm to generate such sets is devised. Using the same diagram, we propose an algorithm to generate efficiently anisotropic triangulations of low-dimensional manifolds embedded in high-dimensional spaces. Our algorithm is provable, but produces disappointing results. Finally, we study Riemannian Voronoi diagrams and introduce discrete Riemannian Voronoi diagrams, which employ recent developments in the numerical computation of geodesic distances and whose computation is accelerated through the use of an underlying anisotropic graph structure. We give conditions that guarantee that our discrete structure is combinatorially equivalent to the Riemannian Voronoi diagram and that its dual is an embedded triangulation, using both straight and curved simplices. We obtain significantly better results than with our other methods, but the overall utility of

Dans ce manuscrit, nous présentons un modèle à effets mixtes, présenté dans un cadre Bayésien, permettant d'estimer la progression temporelle d'un phénomène biologique à partir d'observations répétées, à valeurs dans une variété Riemannienne, et obtenues pour un individu ou groupe d'individus. La progression est modélisée par des trajectoires continues dans l'espace des observations, que l'on suppose être une variété Riemannienne. La trajectoire moyenne est définie par les effets mixtes du modèle. Pour définir les trajectoires de progression individuelles, nous avons introduit la notion de variation parallèle d'une courbe sur une variété Riemannienne. Pour chaque individu, une trajectoire individuelle est construite en considérant une variation parallèle de la trajectoire moyenne et en reparamétrisant en temps cette parallèle. Les transformations spatio-temporelles sujet-spécifiques, que sont la variation parallèle et la reparamétrisation temporelle sont définnies par les effets aléatoires du modèle et permettent de quantifier les changements de direction et vitesse à laquelle les trajectoires sont parcourues. Le cadre de la géométrie Riemannienne permet d'utiliser ce modèle générique avec n'importe quel type de données définies par des contraintes lisses. Une version stochastique de l'algorithme EM, le Monte Carlo Markov Chains Stochastic Approximation EM (MCMC-SAEM), est utilisé pour estimer les paramètres du modèle au sens du maximum a posteriori. L'utilisation du MCMC-SAEM avec un schéma numérique permettant de calculer le transport parallèle est discutée dans ce manuscrit. De plus, le modèle et le MCMC-SAEM sont validés sur des données synthétiques, ainsi qu'en grande dimension. Enfin, nous des résultats obtenus sur différents jeux de données liés à la santé. / We propose a generic Bayesian mixed-effects model to estimate the temporal progression of a biological phenomenon from manifold-valued observations obtained at multiple time points for an individual or group of individuals. The progression is modeled by continuous trajectories in the space of measurements, which is assumed to be a Riemannian manifold. The group-average trajectory is defined by the fixed effects of the model. To define the individual trajectories, we introduced the notion of « parallel variations » of a curve on a Riemannian manifold. For each individual, the individual trajectory is constructed by considering a parallel variation of the average trajectory and reparametrizing this parallel in time. The subject specific spatiotemporal transformations, namely parallel variation and time reparametrization, are defined by the individual random effects and allow to quantify the changes in direction and pace at which the trajectories are followed. The framework of Riemannian geometry allows the model to be used with any kind of measurements with smooth constraints. A stochastic version of the Expectation-Maximization algorithm, the Monte Carlo Markov Chains Stochastic Approximation EM algorithm (MCMC-SAEM), is used to produce produce maximum a posteriori estimates of the parameters. The use of the MCMC-SAEM together with a numerical scheme for the approximation of parallel transport is discussed. In addition to this, the method is validated on synthetic data and in high-dimensional settings. We also provide experimental results obtained on health data.