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This dissertation considers several problems related to Ricci flow, including the existence and behavior of solutions. The first goal is to obtain explicit, coordinate-based descriptions of Ricci flow solutions--especially those corresponding to Ricci solitons--on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott's blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz. Next, we consider solsolitons involving the nilsolitons in the Heisenberg case above. This uses work of Lauret, which characterizes solsolitons as certain extensions of nilsolitons, and work of Will, which demonstrates that the space of solsolitons extensions of a given nilsoliton is parametrized by the quotient of a Grassmannian by a finite group. We determine these spaces of solsoliton extensions of Heisenberg nilsolitons, and we also explicitly describe many-parameter families of these solsolitons in dimensions greater than three. Finally, we explore Ricci flow coupled with harmonic map flow, both as it arises naturally in certain bundle constructions related to Ricci flow and as a geometric flow in its own right. In the first case, we generalize a theorem of Knopf that demonstrates convergence and stability of certain locally R[superscript N]-invariant Ricci flow solutions. In the second case, we prove a version of Hamilton's compactness theorem for the coupled flow, and then generalize it to the category of etale Riemannian groupoids. We also provide a detailed example of solutions to the flow on the three-dimensional Heisenberg group. / text

Σκοπός της εργασίας μας είναι η μελέτη κάποιων αναγωγικών χώρων που παρουσιάζουν ενδιαφέρουσα γεωμετρία. Συγκεκριμένα, μελετάμε τη γεωμετρία της σφαίρας S^n όταν αυτή είναι αμφιδιαφορική με έναν χώρο πηλίκο G/K και την γεωμετρία των πολλαπλοτήτων Stiefel SO(n)/SO(n-k) (το σύνολο όλων των k-πλαισίων του R^n). Ένας ομογενής χώρος αποτελεί επέκταση των ομάδων Lie, καθώς είναι μια λεία πολλαπλότητα M στην οποία δρα μεταβατικά μια ομάδα Lie G. Κάθε τέτοιος χώρος δίνεται ως M = G/K, όπου K = {g\in G : gp = p} (p \in M). Η βασική γεωμετρική ιδιότητα των ομογενών χώρων είναι ότι αν γνωρίζουμε την τιμή κάποιου γεωμετρικού μεγέθους σε ένα σημείο του χώρου, τότε μπορούμε να υπολογίσουμε την τιμή του μεγέθους αυτού σε οποιοδήποτε άλλο σημείο. Το ιδιαίτερο χαρακτηριστικό των αναγωγικών χώρων G/K είναι ότι υπάρχει ένας Ad(K)-αναλλοίωτος υπόχωρος της άλγεβρας Lie(G). Η περιγραφή όλων των μεταβατικών δράσεων μιας ομάδας Lie σε μια πολλαπλότητα M αποτελεί ένα δύσκολο πρόβλημα. Για την περίπτωση των σφαιρών αυτές έχουν περιγραφτεί το 1953 από τους Montgomery-Samelson-Borel. Στην εργασία μας μελετάμε τη γεωμετρία (καμπυλότητες, μετρικές Einstein) των σφαιρών S^3, S^5 όταν αυτές είναι αμφιδιαφορικές με τα πηλίκα S^3 = SO(4)/SO(3) = SU(2) και S^5 = SO(6)/SO(5) = SU(3)/SU(2). Αντίστοιχα προβλήματα εξετάζονται για τις πολλαπλότητες Stiefel SO(n)/SO(n-k), όπου η περιγραφή όλων των SO(n)-αναλλοίωτων μετρικών παρουσιάζει δυσκολία, λόγω του ότι η ισοτροπική αναπαράστασή τους περιέχει ισοδύναμα υποπρότυπα. Μελετάμε για ποιές από τις συγκεκριμένες πολλαπλότητες η μετρική που επάγεται από τη μορφή Killing είναι μετρική Einstein και περιγράφουμε αναλυτικά τις διαγώνιες SO(n)-αναλλοίωτες μετρικές Einstein στις πολλαπλότητες SO(n)/SO(n-2). Επιπλέον παρουσιάζουμε και ένα καινούργιο αποτέλεσμα, ότι στην πολλαπλότητα SO(5)/SO(2) οι μοναδικές SO(5)-αναλλοίωτες μετρικές Einstein είναι οι μετρικές που είχαν βρεθεί από τον Jensen το 1973. / The purpose of our work is to study homogeneous spaces that present interesting geometry. These include the geometry of the sphere S^n diffeomorphic to a quotient space G/K and the geometry of Stiefel manifolds SO(n)/SO(n-k) (the set of all k-planes in R^n). A homogeneous space is a smooth manifold M in which a Lie group acts transitively. Any such space is given as M = G/K where K = {g\in G : gp = p} (p\in M). The basic geometric property of homogeneous space is that if we know the value of a geometrical object at a point of the space, then we can estimate the value of thiw quantity at any other point. The special feature of reductive homogeneous space G/K is that there exists an Ad(K)-invariant subspace of the Lie algebra Lie(G). The description of all transitive actions of a Lie group into a manifold M is a difficult problem. In the case of spheres such actions have been described in 1953 by the Montgomery, Samelson and Borel. In our work we study the geometry (curvature, Einstein metrics) of the sphere S^3 = SO(4)/SO(3) = SU(2), S^5 = SO(6)/SO(5) = SU(3)/SU(2). Similar problems are examined for the Stiefel manifolds SO(n)/SO(n-k). The description of all SO(n)-invariant metrics presents serious difficulties because the isotropy representation contains equivalent submodules. We study for which of the manifolds SO(n)/SO(n-k) the metric induced by the Killing form is an Einstein metric and we describe in detail the diagonal SO(n)-invariant Einstein metrics on the Stiefel manifolds SO(n)/SO(n-2). In addition, we give the new result that for the Stiefel manifold SO(5)/SO(2) the unique SO(5)-invariant Einstein metrics are the metrics found by Jensen in 1973.

In this work we study the geometric aspects of Lie groups from the view point of the Riemannian geometry, by means of invariant geometric structures associated. We present some properties on curvatures of metrics left invariants and bi-invariant one on Lie groups. We also present a treatment of the Lie algebras unimodular, including the tridimensional case. Most of the results studied are from the article of John Milnor: Curvatures of Left Invariant Metrics on Lie Groups. / Neste trabalho estudamos os aspectos geometricos de grupos de Lie, do ponto de vista da geometria Riemanniana, por meio de estruturas geometricas invariantes associadas. Nos apresentamos algumas propriedades de curvaturas com metricas invariante a esquerda e aquelas bi-invariantes em grupos de Lie. Apresentamos tambem um tratamento das algebras de Lie unimodulares, incluindo o caso tridimensional. A maioria dos resultados estudados foram retirados do artigo de John Milnor: Curvatures of Left Invariant Metrics on Lie Groups. / Mestre em Matemática

Soit μ une mesure de probabilité borélienne sur SL m+1 (R) tel que le sous-groupe engendré par le support de μ est Zariski dense. Soit V une représentation irréductible de dimension finie de SL m+1 (R). D’après un théorème de Furstenberg, il existe une unique mesure μ-stationnaire sur PV et nous nous somme intéressés à la décroissance de Fourier de cette mesure. Le résultat principal de cette thèse est que la transformée de Fourier de la mesure stationnaire a une décroissance polynomiale. À partir de ce résultat, nous obtenons un trou spectral de l’opérateur de transfert, dont les propriétés nous permettent d’établir un terme d’erreur exponentiel pour le théorème de renouvellement dans le cadre des produits de matrices aléatoires. L’ingrédient essentiel est une propriété de décroissance de Fourier des convolutions multiplicatives de mesures sur R n , qui est une généralisation d’un théorème de Bourgain en dimension 1. Nous établissons cet ingrédient en utilisant un estimée somme produit de He et de Saxcé.Dans la dernière partie, nous généralisons un résultat de Lax et Phillips et un résultat de Hamenstädt sur la finitude des petites valeurs propres de l’opérateur de Laplace sur les variétés hyperboliques géométriquement finies. / Let μ be a Borel probability measure on SL m+1 (R), whose support generates a Zariski dense subgroup. Let V be a finite dimensional irreducible linear representation of SL m+1 (R). A theorem of Furstenberg says that there exists a unique μ-stationary probability measure on PV and we are interested in the Fourier decay of the stationary measure. The main result of the thesis is that the Fourier transform of the stationary measure has a power decay. From this result, we obtain a spectral gap of the transfer operator, whose properties allow us to establish an exponential error term for the renewal theorem in the context of products of random matrices. A key technical ingredient for the proof is a Fourier decay of multiplicative convolutions of measures on R n , which is a generalisation of Bourgain’s theorem on dimension 1. We establish this result by using a sum-product estimate due to He-de Saxcé. In the last part, we generalize a result of Lax-Phillips and a result of Hamenstädt on the finiteness of small eigenvalues of the Laplace operator on geometrically finite hyperbolic manifolds

Dans cette thèse, nous proposons de nouvelles representations pourles poses du mouvement humain, apprises sur des données réelles, envue d’une synthèse de nouveaux mouvements en temps-réel. Dans unepremière partie, nous exploitons une méthode statistique adaptée auxgroupes de Lie (Analyse en Géodésiques Principales, AGP) pour approximerla variété des poses d’un sujet en mouvement, à partir de donnéesde capture de mouvement. Nous proposons un algorithme de cinématiqueinverse exploitant cette paramétrisation réduite, permettantpar construction de synthétiser des poses proches des données initiales.Nous validons ce modèle cinématique par une application à la compressionde données de mouvements, dans laquelle seules quelques trajectoiresdes extrémités des membres du squelettes permettent de reconstruireune bonne approximation de l’ensemble des données initiales.Dans une deuxième partie, nous étendons cette approche à l’animationphysique de personnages virtuels. La paramétrisation réduitepar AGP fournit les coordonnées généralisées de la formulation Lagrangiennede la mécanique. Nous dérivons un intégrateur temporelexplicite basé sur les intégrateurs variationnels. Afin d’en améliorer lastabilité, nous proposons un modèle d’amortissement inspiré de l’algorithmede Levenberg-Marquardt. Nous présentons également une méthodegéométrique d’apprentissage des limites angulaires sur des donnéesde capture de mouvement, ainsi que leur application comme contraintescinématiques.Dans une troisième partie, nous abordons le problème du contrôledu mouvement. En formulant les étapes de la simulation physique d’unepart, et de la cinématique inverse d’autre part comme deux programmesquadratiques, nous proposons un algorithme de pseudo-contrôle parinterpolation des métriques, permettant un compromis intuitif entre simulationphysique non-contrôlée, et cinématique inverse. Cette approchefaisant intervenir des forces externes, nous proposons une formulationalternative, utilisant uniquement les forces associées à la paramétrisationréduite des poses. Cette formulation est obtenue par relaxationdu problème théorique de contrôle sous contraintes unilatérales, nonconvexe,en un programme quadratique convexe. Ces algorithmes sontévalués sur des contrôleurs d’équilibre et de suivi. / In this thesis, we propose novel, data-driven representations for humanposes, suitable for real-time synthesis of novel character motion. Inthe first part, we exploit Lie group statistical analysis techniques (PrincipalGeodesic Analysis, PGA) to approximate the pose manifold of amotion capture sequence by a reduced set of pose geodesics. We proposean inverse kinematics algorithm using this reduced parametrizationto automatically produce poses that are close to the learning set. Wedemonstrate the efficiency of the resulting pose model by an applicationto motion capture data compression, where only a few end-effector trajectoriesare used to recover a good approximation of the initial data.In the second part, we extend this approach to the physically-basedanimation of virtual characters. The PGA-reduced parametrization providesgeneralized coordinates in a Lagrangian formulation of mechanics.We derive an explicit time integrator by approximating existingvariational integrators, and propose a damping model based on theLevenberg-Marquardt algorithm. We also describe a geometric, datadriven,angular limit learning algorithm, and the associated kinematicconstraints.In the third part, we reach the problem of task-space motion control.By formulating both physical simulation and inverse kinematicstime stepping schemes as two quadratic programs, we propose a simplepseudo-control algorithm that interpolates between the two metrics.This allows for an intuitive trade-off between uncontrolled simulationand kinematic manipulation. Since this approach makes use of externalforces, we propose an alternate formulation using only the generalizedforces associated to the pose parametrization. A control algorithmis obtained by the relaxation of the exact, non-convex control problemunder unilateral constraints, into a convex quadratic program. Thesealgorithms are evaluated on simple balance and tracking controllers.

We will show that the only connected Euclidean spheres admitting a structure of Lie group are S1 and S3, for all n greater than or equal to 1. We will do this through the study of properties of the De Rham cohomology groups of sphere Sn and of compact connected Lie groups. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Mostraremos que as únicas esferas euclidianas conexas que admitem uma estrutura de grupo de Lie são S1 e S3, para todo n maior ou igual a 1. Faremos isso por intermédio do estudo de propriedades dos grupos de cohomologia de De Rham das esfereas Sn e dos grupos de Lie compactos e conexos.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Nesta dissertação, descrevemos resultados obtidos por Hilário Alencar e A. Gervasio Colares, publicado no Annals of Global Analysis and Geometry em 1998. Inicialmente, obtemos fórmulas integrais para a curvatura r-média, as quais generalizam fórmulas de Minkowski. Além disso, usando estas fórmulas, caracterizamos as hipersuperfícies compactas imersas no espaço Euclidiano, esférico ou hiperbólico cujo conjunto de pontos nestes espaços que não pertencem as hipersuperfícies totalmente geodésicas tangentes às hipersuperfícies compactas é aberto e não vazio. Outrossim, obtemos ainda resultados relacionados com a estabilidade. As demonstrações destes resultados são obtidas através da fórmula integral de Dirichlet para o operador linearizado da curvatura r-média de uma hipersuperfície imersa no espaço Euclidiano, esférico ou hiperbólico, bem como do uso de um resultado recente provado por Hilário Alencar, Walcy Santos e Detang Zhou no preprint Curvature Integral Estimates for Complete Hypersurfaces. Ressaltamos que esta dissertação foi baseada na versão corrigida por Hilário Alencar do artigo publicado no Annals of Global Analysis and Geometry.

Orientador: Pedro José Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:06:57Z (GMT). No. of bitstreams: 1 Franca_ElizeuCleberdosSantos_M.pdf: 3490670 bytes, checksum: 6589d1bb5b880e51625fb93abb2964a9 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho, apresentaremos a teoria básica de simetrias de equações diferenciais, focando na busca por soluções invariantes de operadores diferenciais definidos em fibrados vetoriais com relação a ação transversal de um grupo de Lie no fibrado em questão / Abstract: In this work we will give the basic theory of symmetries of differential equations. The goal of this work is searching for invariant solutions of differential operators which are defined on vector bundles with respect to the transverse action of a Lie group in such bundle / Mestrado / Matematica / Mestre em Matemática