Estudamos exemplos de métricas Finsler simétricas e não-simétricas em S^n, CP^n e HP^n com uma quantidade finita de geodésicas fechadas ou com uma quantidade pequena de geodésicas fechadas \"curtas\". São os chamados exemplos de Katok. Usamos como referência o artigo \"Geometry of the Katok examples\" de Wolfgang Ziller. Verificamos que existem métricas Finsler cujo número de geodésicas fechadas é 2n (no caso de S^ e S^), n(n+1) (no caso de CP^n) e 2n(n+1) (no caso de CP^n). Tais exemplos são construídos numa vizinhança qualquer da métrica Riemanniana canônica dessas variedades. / We study examples of symmetric and non-symmetric Finsler metrics on S^n, CP^n and HP^n with a finite number of closed geodesics or with a small number of \"short\" closed geodesics. These are the well known Katok\'s examples. We use Ziller\'s article Geometry of the Katok examples. We exhibit Finsler metrics whose number of closed geodesics is 2n (in the case of S^ and S^), n(n+1) (in the case of CP^n) and 2n(n+1) (in the case of HP^n). Such examples are found in any neighborhood of the canonical Riemannian metric on these manifolds.

Estudamos exemplos de métricas Finsler simétricas e não-simétricas em S^n, CP^n e HP^n com uma quantidade finita de geodésicas fechadas ou com uma quantidade pequena de geodésicas fechadas \"curtas\". São os chamados exemplos de Katok. Usamos como referência o artigo \"Geometry of the Katok examples\" de Wolfgang Ziller. Verificamos que existem métricas Finsler cujo número de geodésicas fechadas é 2n (no caso de S^ e S^), n(n+1) (no caso de CP^n) e 2n(n+1) (no caso de CP^n). Tais exemplos são construídos numa vizinhança qualquer da métrica Riemanniana canônica dessas variedades. / We study examples of symmetric and non-symmetric Finsler metrics on S^n, CP^n and HP^n with a finite number of closed geodesics or with a small number of \"short\" closed geodesics. These are the well known Katok\'s examples. We use Ziller\'s article Geometry of the Katok examples. We exhibit Finsler metrics whose number of closed geodesics is 2n (in the case of S^ and S^), n(n+1) (in the case of CP^n) and 2n(n+1) (in the case of HP^n). Such examples are found in any neighborhood of the canonical Riemannian metric on these manifolds.

<div>In the present work, we consider a class of Finsler metrics using the warped product notion introduced by B. Chen, Z. Shen and L. Zhao (2018), with another “warping”, one that is consistent with the form of metrics modeling static spacetimes and simplified by spherical symmetry over spatial coordinates, which emerged from the Schwarzschild metric in isotropic coordinates. We will give the PDE characterization for the proposed metrics to be Ricci-flat and construct explicit examples. Whenever possible, we describe both positive-definite solutions and solutions with Lorentz signature. For the latter, the 4-dimensional metrics may also be studied as Finsler spacetimes.</div>

<p>The curvatures in Finsler geometry can be defined in similar ways as in Riemannian geometry. However, since there are fewer restrictions on the metrics, many geometric quantities arise in Finsler geometry which vanish in the Riemannian case. These quantities are generally known as non-Riemannian quantities and interact with the curvatures in controlling the global geometrical and topological properties of Finsler manifolds. In the present work, we study general weighted Ricci curvatures which combine the Ricci curvature and the S-curvature, and define a weighted flag curvature which combines the flag curvature and the T -curvature. We characterize Randers metrics of almost isotropic weighted Ricci curvatures and show the general weighted Ricci curvatures can be divided into three types. On the other hand, we show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, generalizing the Gromoll-Meyer theorem in Riemannian geometry.</p>

Transnormal functions are generalization of distance functions and this topic has some applications in Physics and real world problems. In this work, some results are generalized from Riemannian case to the Finsler one. Moreover certain new phenomena that happen only in Finsler spaces are discussed. To have a better understanding, certain examples based on the mentioned results in Randers spaces are provided. Moreover, some applications on propagation of waves of fire and water are introduced / As funções transnormais são a generalização da função de distância e este tópico tem algumas aplicações em Física e no mundo real. Neste trabalho, alguns resultados do caso riemanniana para o Finsler são generalizados. Alem disso, alguns fenômenos novos que ocorrem apenas nos espaços de Finsler são discutidos. Para ter uma melhor compreensão, são fornecidos certos exemplos com base nos resultados mencionados nos espaços de Randers. Além disso, algumas aplicações sobre propagação de ondas de fogo e água são introduzidas.

Transnormal functions are generalization of distance functions and this topic has some applications in Physics and real world problems. In this work, some results are generalized from Riemannian case to the Finsler one. Moreover certain new phenomena that happen only in Finsler spaces are discussed. To have a better understanding, certain examples based on the mentioned results in Randers spaces are provided. Moreover, some applications on propagation of waves of fire and water are introduced / As funções transnormais são a generalização da função de distância e este tópico tem algumas aplicações em Física e no mundo real. Neste trabalho, alguns resultados do caso riemanniana para o Finsler são generalizados. Alem disso, alguns fenômenos novos que ocorrem apenas nos espaços de Finsler são discutidos. Para ter uma melhor compreensão, são fornecidos certos exemplos com base nos resultados mencionados nos espaços de Randers. Além disso, algumas aplicações sobre propagação de ondas de fogo e água são introduzidas.

Une approche fondamentale pour la résolution de problémes de contrôle optimal est basée sur le principe de programmation dynamique. Ce principe conduit aux équations d'Hamilton-Jacobi, qui peuvent être résolues numériquement par des méthodes classiques comme la méthode des différences finies, les méthodes semi-lagrangiennes, ou les schémas antidiffusifs. À cause de la discrétisation de l'espace d'état, la dimension des problèmes de contrôle pouvant être abordés par ces méthodes classiques est souvent limitée à 3 ou 4. Ce phénomène est appellé malédiction de la dimension. Cette thèse porte sur les méthodes numériques max-plus en contôle optimal deterministe et ses analyses de convergence. Nous étudions et developpons des méthodes numériques destinées à attenuer la malédiction de la dimension, pour lesquelles nous obtenons des estimations théoriques de complexité. Les preuves reposent sur des résultats de théorie de Perron-Frobenius non linéaire. En particulier, nous étudions les propriétés de contraction des opérateurs monotones et non expansifs, pour différentes métriques de Finsler sur un cône (métrique de Thompson, métrique projective d'Hilbert). Nous donnons par ailleurs une généralisation du "coefficient d'ergodicité de Dobrushin" à des opérateurs de Markov sur un cône général. Nous appliquons ces résultats aux systèmes de consensus ainsi qu'aux équations de Riccati généralisées apparaissant en contrôle stochastique.

En 1949, C. Loewner a demontré dans un travail non publié l'inégalité systolique optimale du tore T reliant l'aire au carré de la systole. Par la systole on désigne la longueur du plus court lacet non contractile de T. De plus, l' égalité est atteinte si et seulement si le tore est plat hexagonal. Ce résultat a donné naissance à la géométrie systolique. Dans cette thèse, nous étudions des inégalités de type systolique portant sur les longueurs minimales de différentes courbes et pas seulement la systole.Dans un premier temps, nous démontrons trois inégalités géométriques optimales conformes sur la bouteille de Klein reliant l'aire au produit des longueurs des plus courts lacets noncontractiles dans des classes d'homotopie libres différentes. Pour chaque classe conforme, nous décrivons la métrique extrémale réalisant le cas d'égalité.Nous établissons ensuite des inégalités géométriques optimales sur le ruban deMobius muni d'une métrique de Finsler. Ces inégalités géométriques relient la systole et la hauteur du ruban de Mobius à son volume de Holmes-Thompson. Nous en déduisons une inégalité systolique optimale sur la bouteille de Klein munie d'une métrique de Finsler avec des symétries. Nous décrivons également une famille de métriques extrémales dans les deux cas.Dans le troisième travail, nous démontrons une inégalité systolique critique sur la surface de genre deux. Plus précisément, il est connu que la surface de genre deux admet une métrique Riemannienne plate à singularités coniques qui est extrémale parmi les métriques à courbure nonpositive pour l' inégalité systolique. Nous montrons que cette métrique est en fait critique pour des variations lentes de métriques, cette fois-ci sans hypothèse de courbure, pour un autre problème systolique portant sur les longueurs des plus courts lacets non contractiles dans certaines classes d'homotopie libres données. Ces classes d'homotopie correspondent aux lacets systoliques et deux-systoliques de la surface extrémale / In 1949, C. Loewner proved in an unpublished work that the two-torus T satisfies an optimal systolic inequality relating the area of the torus to the square of its systole. By a systole here we mean the smallest length of a noncontractible loop in T. Furthermore, the equality is attained if and only if the torus is flat hexagonal. This result led to whatwas called later systolic geometry. In this thesis, we study several systolic-like inequalities. These inequalities involve the minimal length of various curves and not merely the systole.First we obtain three optimal conformal geometric inequalities on Riemannian Klein bottles relating the area to the product of the lengths of the shortest noncontractible loops in different free homotopy classes. We describe the extremal metrics in each conformal class.Then we prove optimal systolic inequalities on Finsler Mobius bands relating the systoleand the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimalsystolic inequality for Finsler Klein bottles with symmetries. We describe extremal metric families in both cases.Finally, we prove a critical systolic inequality on genus two surface. More precisely, it is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved Riemannian metrics. We show that this piecewise flat metric is also critical for slow metric variations, this time without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes. The free homotopy classes considered correspond to those of the systolic loops and the second-systolic loops of the extremal surface

The first aim of this PhD Thesis is devoted to the study of geodesic distances defined on a subdomain of a Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. Then one shows that the uniform convergence, on compact sets, of these distances can be equivalently characterized in terms of Gamma-convergence of several kinds of variational problems. Moreover, it investigates the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. The second purpose is to obtain the integral representation of some classes of local functionals, depending on a family of vector fields, that satisfy a weak structure assumption. These functionals are defined on degenerate Sobolev spaces and they are assumed to be not translations-invariant. Then one proves some Gamma-compactness results with respect to both the strong topology of L^p and the strong topology of degenerate Sobolev spaces.

Made available in DSpace on 2014-07-29T16:02:22Z (GMT). No. of bitstreams: 1 dissertacao karoline fernandes.pdf: 700169 bytes, checksum: bbcf93fe91f369b6605215c70576e124 (MD5) Previous issue date: 2010-02-26 / We will study the Finsler metric, on a manifold M, defined as the sum of a Riemannian metric and a 1-form, they are known as Randers metric. We will classify those that are locally dually flat, that is, for all point exists a coordinate system in which the equation of the geodesic has a special form, the coefficients of spray is given in terms of the metric one and a local scalar function, we will also characterize the Randers metric that is locally dually flat with almost isotropic flag curvature / Estudaremos as métricas de Finsler, em uma variedade M, definidas como soma de uma métrica Riemanniana e de uma 1-forma, elas são conhecidas como métricas de Randers. Classificaremos aquelas que são localmente dualmente flat, isto é, para todo ponto existe um sistema de coordenadas no qual a equação das geodésicas tem uma forma especial pois os coeficientes do spray são dados em termos da métrica e de uma função escalar, caracterizaremos também as métricas de Randers que são localmente dualmente flat com curvatura flag quase-isotrópica