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The focus of this thesis is on the investigation of the geodesics of ruled surfaces.

There are two main approaches to solve the problem of finding closed geodesics on a Riemannian manifold M. The variational approach views a closed geodesic as a closed curve which happens to be a geodesic and it looks for critical points of the energy functional, while the dynamical systems approach views a closed geodesic as a geodesic which happens to close up and looks for periodic orbits of the geodesic ow on the unit tangent bundle.

We construct products on the homology of quotients by finite group actions of the free loop space ΛM of a compact manifold M. We compute some of the these products in the case M is as sphere. We show that there are nonnilpotent classes with respect to these products for spheres. The energy functional on ΛM associated to a Riemannian metric on M is invariant under the group actions we consider. We therefore retain information about geometrically distinct closed geodesics.

<p>Existence of closed geodesics on compact manifolds was first proved by Lyusternik and Fet in the 1950s using Morse theory, and the corresponding problem for orbifolds was studied by Guruprasad and Haefliger, who proved existence of a closed geodesic of positive length in numerous cases. In this thesis, we develop an alternative approach to the problem of existence of closed geodesics on compact orbifolds by studying the geometry of group actions. We give an independent and elementary proof that recovers and extends the results of Guruprasad and Haefliger for developable orbifolds. We show that every compact orbifold of dimension 2, 3, 5 or 7 admits a closed geodesic of positive length, and we give an inductive argument that reduces the existence problem to the case of a compact developable orbifold of even dimension whose singular locus is zero-dimensional and whose orbifold fundamental group is infinite torsion and of odd exponent. Stronger results are obtained under curvature assumptions. For instance, one can show that infinite torsion groups do not act geometrically on simply connected manifolds of nonpositive or nonnegative curvature, and we apply this to prove existence of closed geodesics for compact orbifolds of nonpositive or nonnegative curvature. In the general case, the problem of existence of closed geodesics on compact orbifolds is seen to be intimately related to the group-theoretic question of finite presentability of infinite torsion groups, and we explore these and other properties of the orbifold fundamental group in the last chapter.</p> / Doctor of Philosophy (PhD)

Cette thèse porte sur certaines familles à un paramètre de surfaces de Riemann compactes de genre 2 définies par des surfaces de translation. Les familles que nous considérons constituent des géodésiques de Teichmüller dans l'espace des modules.Nous nous attachons en particulier à décrire ces surfaces par leurs matrices des périodes et par les équations des courbes algébriques associées.Nous étudions notamment les automorphismes admissibles par les surfaces de certaines de ces familles.Le principal résultat consiste en une caractérisation explicite des matrices des périodes des courbes réelles à trois composantes réelles appartenant à la famille obtenue par projection dans l'espace des modules de la SL(2,R)-orbite de la surface de translation en «L» pavée par trois carreaux.Nous montrons enfin, grâce à une interprétation en termes de transformations de Schwarz-Christoffel, comment calculer numériquement une équation de la courbe algébrique définie par une surface de translation en «L». / In this thesis we study some one parameter families of compact Riemann surfaces of genus 2 defined by translation surfaces.The families we consider are Teichmüller geodesics in the moduli space.We mainly describe these surfaces by means of period matrices and equations of the associated algebraic curves.We study admissible automorphisms for surfaces in some of those families.The main result is an explicit characterisation of period matrices of real curves with three real components belonging to the family obtained by projecting the SL(2,R)-orbit of the «L»-shaped translation surface tiled by three squares into the moduli space.We finally show, using an interpretation in terms of Schwarz-Christoffel transformations, how to numerically compute an equation of the algebraic curve defined by a «L»-shaped translation surface.

Orientador: Sueli Irene Rodrigues Costa / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-09-11T21:20:45Z (GMT). No. of bitstreams: 1 Bressan_JoaoPaulo_D.pdf: 1164660 bytes, checksum: 4825edafe6fbea5e3bc43934bc528376 (MD5) Previous issue date: 2012 / Resumo: A partir das desigualdades de Hamming e Gilbert-Varshamov obtém-se um limitante superior e um limitante inferior para o número de pontos de um código numa variedade flag geométrica. Isto é feito tomando-se uma estimativa para o volume de bolas geodésicas, que resulta de cálculos envolvendo a curvatura seccional destas variedades. Em particular, são derivados limitantes para empacotamentos de esferas numa variedade de Grassmann complexa. Um limitante superior para a distância mínima também é obtido através da inversa da função que calcula o volume de um chapéu esférico. Esta técnica geométrica também é aplicada no estudo de limitantes para empacotamentos em alguns casos particulares de variedades flag maximais. Através de procedimentos computacionais, tais limitantes são implementados numericamente em alguns exemplos. Uma motivação para este trabalho foi à busca de possíveis extensões de alguns resultados sobre as grassmanianas complexas, cujo interesse na área de comunicações vem de uma interpretação que pode ser feita da transmissão em canais MIMO não coerentes via códigos em tais variedades / Abstract: Upper and lower bounds for the number of points of codes in geometric flag manifolds are obtained from Hamming and Gilbert-Varshamov inequalities. This is done by taking an estimate for the volume of geodesic balls, as a result of calculations involving the sectional curvature of such manifolds. As a particular case, sphere packing bounds in complex Grassmann manifolds are derived. An upper bound on the minimum distance is also obtained through the inverse mapping for the volume of spherical caps. This geometric technique is also applied in the study of sphere packing bounds in some particular cases of full-flag manifolds. Such bounds are numerically implemented in some examples. One motivation for this work was the search for possible extensions of some results on complex Grassmann manifolds, which interest in communications comes from a model for the transmition on non-coherent MIMO channels via codes in such manifolds / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

The motion of neutral test particles, both massive and massless, in the space time of a charged source described by the Reissner-Nordström solution is studied. This solution is characterized by two parameters, mass and charge, which defines the horizons of the source. When the mass is larger than the charge, the solution describes a black hole, with two distinct horizons. When the mass and charge are equal there is an extremal black hole, and both horizons merge to one. Finally, when the charge is larger than the mass there is a naked singularity, with no horizon. The structure and properties of these different type of solution are presented and discussed. A general solution of the equations of motion is presented in function of the Weierstrass elliptic function &weierp;. In addition, the possible orbits for test particles are discussed, and the conditions for existence of closed, circular or escape orbits are presented. The classifications is made based on the particles energy, and the mass and charge of the source. We find that all mentioned orbits are allowed for the three different type of solutions. In particular, for extremal black holes and naked singularities, we find stable circular orbits located outside the event horizon and hence being visible for an external observer. / O movimento de partículas teste neutras, ambas massivas e sem massa, no espaço-tempo de uma fonte carregada descrita pela solução de Reissner-Nordström é estudada. Essa solução é caracterizada por dois parâmetros, massa e carga, que definem os horizontes da fonte. Quando a massa é maior que a carga tal solução descreve um buraco negro com dois horizontes distintos. Quando a massa e a carga são iguais há um buraco negro extremo, e ambos os horizontes se unem em um. Finalmente, quando a carga é maior que a massa, há uma singularidade nua, sem horizontes. A estrutura e as propriedades dessas diferentes soluções são apresentadas e discutidas. Uma solução geral da equação de movimento é apresentada em termos da função elíptica de Weierstrass, &weierp;. Além do mais as possiveis órbitas para uma partícula teste são discutidas, e as condições para existência de órbitas fechadas, circulares e de escape são apresentadas. A classificação é feita a partir da energia da partícula, e da massa e carga da fonte. Encontramos que todas as orbitas mencionadas são permitidas nos três diferentes tipos de soluções. Em partícular, para buracos negros extremos e singularidades nuas, encontramos órbitas circulares estáveis localizadas fora do horizonte de eventos e, consequentemente, sendo visível para observadores externos.

Orientador: Caio José Colletti Negreiros / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T12:45:02Z (GMT). No. of bitstreams: 1 Grama_LinoAndersondaSilva_D.pdf: 1119551 bytes, checksum: d2dc2c993629f40f7976e91497c5d219 (MD5) Previous issue date: 2011 / Resumo: O principal objetivo deste trabalho é o estudo de aplicações harmônicas em variedades flag generalizadas. Na primeira parte do trabalho, consideramos aplicações cujo domínio é uma superfície de Riemann. Provamos que toda aplicação holomorfa-horizontal na variedade flag é uma aplicação equiharmônica (ie, harmônica com respeito a cada métrica invariante na variedade flag). Obtemos também as fórmulas de Plucker para curvas holomorfa-horizontais na variedade flag maximal. Na segunda parte do trabalho, consideramos aplicações harmônicas cujo domínio possui dimensão 1 ( ie, geodésicas) na variedade flag. Provamos que toda variedade ag generalizada admite curvas que são geodésicas com respeito a cada métrica invariante. Tais curvas são chamadas equigeodésicas. Fornecemos uma descrição algébrica para tais curvas e exibimos famílias de equigeodésicas em diversas famílias de variedades flag / Abstract: The main goal of this work is the study of harmonic maps in generalized flag manifolds. In the first part of the work, we consider maps whose domain is a Riemann surface. We prove that every holomorphic-horizontal map in the flag manifold is an equiharmonic map (i.e. harmonic with respect to each invariant metric in the flag manifold). We also obtain the Plucker formulae for holomorphic-horizontal curves in full flag manifolds. In the second part of the work, we consider harmonic maps whose domain has dimension one (i.e. geodesics) in the ag manifold. We prove that every generalized flag manifold admit curves that are geodesics with respect to each invariant metric. Such curves are called equigeodesics. We provide an algebraic characterization for such curves and exhibit families of equigeodesics in several families of flag manifolds / Doutorado / Doutor em Matemática

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-29T13:28:57Z No. of bitstreams: 1 douglasferreirabrum.pdf: 632780 bytes, checksum: 1da883a558292ba219387c3fdf6f98af (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-29T19:37:42Z (GMT) No. of bitstreams: 1 douglasferreirabrum.pdf: 632780 bytes, checksum: 1da883a558292ba219387c3fdf6f98af (MD5) / Made available in DSpace on 2017-05-29T19:37:42Z (GMT). No. of bitstreams: 1 douglasferreirabrum.pdf: 632780 bytes, checksum: 1da883a558292ba219387c3fdf6f98af (MD5) Previous issue date: 2013-08-30 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Esse trabalho visa descrever o espaço de Módulos de m-uplas geodésicas complexas distintas em H2c nos casos regular, especial e degenerado. Para tal fim faremos uso da matriz de Gram e dos invariantes (d-invariantes, δ-invariantes, invariante angular e invariantes parabólicos) que descrevem unicamente a classe de congruência de PU(2, 1) de m-uplas ordenadas de geodésicas complexas distintas nos diferentes casos supracitados. / This work aims to describe the Modules space of m-tuples distinct complex geodesics in H2c in the cases regular, special and degenerate. To this end we use the Gram matrix and the invariant (d-invariant, δ-invariants, angular invariant and parabolic invariants) that define uniquely the PU(2,1)-congruence class of ordered m-uplas of distinct complex geodesics in the different cases above.