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Variedades de Einstein e Ricci solitons com estrutura de produto torcido / Einstein manifolds and Ricci solitons with warped product structureSousa, Márcio Lemes de 03 July 2015 (has links)
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Previous issue date: 2015-07-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, primarily, we studied warped products semi-Riemannian Einstein manifolds.
We considered the case in that the base is conformal to an n-dimensional pseudo-
Euclidean space and invariant under the action of an (n 1)-dimensional translation
group. We constructed new examples of Einstein warped products with zero Ricci curvature
when the fiber is Ricci-flat. In particular, we obtain explicit solutions, in the case
vacuum, for Einstein field equation. Furthermore, we consider M = B f F warped product
gradient Ricci solitons. We proved that the potential function depends only on the
base and the fiber F is necessarily Einstein manifold. We provide all such solutions in
the case of steady gradient Ricci solitons when the base is conformal to an n-dimensional
pseudo-Euclidean space, invariant under the action of an (n1)-dimensional translation
group, and the fiber F is Ricci-flat. / Nesta tese, primeiramente, estudamos variedades produto torcido semi-Riemannianas de
Einstein, considerando-se o caso em que a base é conforme ao espaço pseudo- Euclidiano
n -dimensional e invariante sob a ação de um grupo de translações (n1)-dimensional.
Construímos novos exemplos de métricas produto torcido Einstein com curvatura de Ricci
zero quando a fibra é Ricci -flat. Em particular, obtemos soluções explícitas, no caso
de vácuo, para a equação de campo de Einstein. Em seguida, provamos que quando a
variedade M = B f F é um Ricci soliton gradiente a função potencial depende apenas
da base e a fibra F é necessariamente uma variedade de Einstein. Fornecemos todas as
soluções, no caso de Ricci soliton gradiente steady, quando a base é conforme ao espaço
pseudo- Euclidiano n -dimensional, invariante sob a ação de um grupo translações (n1)
- dimensional, e a fibra F é Ricci -flat.
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Géométrie à l'infini de certaines variétés riemanniennes non-compactes / Geometry at infinity of some noncompact Riemannian manifoldsDeruelle, Alix 23 November 2012 (has links)
On s'intéresse à la géométrie globale et asymptotique de certaines variétés riemanniennes non compactes. Dans une première partie, on étudie la topologie et la géométrie à l'infini des variétés riemanniennes à courbure (de Ricci) positive ayant un rapport asymptotique de courbure fini. On caractérise le cas non effondré via la notion de cône asymptotique et on donne des conditions suffisantes sur le groupe fondamental pour garantir un non effondrement. La seconde partie est dédiée à l'étude des solutions de Type III du flot de Ricci à courbure positive et aux solitons gradients de Ricci expansifs (points fixes de Type III) présentant une décroissance quadratique de la courbure. On montre l'existence et l'unicité des cônes asymptotiques de tels points fixes. On donne également des conditions suffisantes de nature algébrique et géométrique pour garantir une géométrie de révolution de tels solitons. Dans une troisième partie, on caractérise la géométrie des solitons gradients de Ricci stables à courbure positive et à croissance volumique linéaire. Puis, on s'intéresse au non effondrement des variétés riemanniennes de dimension trois à courbure de Ricci positive ayant un rapport asymptotique de courbure fini. / We study the global and asymptotic geometry of non-compact Riemannian manifolds. First, we study the topology and geometry at infinity of Riemannian manifolds with nonnegative (Ricci) curvature and finite asymptotic curvature ratio. We focus on the non-collapsed case with the help of asymptotic cones and we give sufficient conditions on the fundamental group to guarantee non-collapsing. The second part is dedicated to the study of (non-negatively curved) Type III Ricci flow solutions. We mainly analyze the asymptotic geometry of Type III self-similar solutions (expanding gradient Ricci soliton) with finite asymptotic curvature ratio. We prove the existence and uniqueness of their asymptotic cones. We also give algebraic and geometric sufficient conditions to guarantee rotational symmetry of such metrics. In the last part, we characterize the geometry of steady gradient Ricci solitons with nonnegative sectional curvature and linear volume growth. Finally, we study the non-collapsing of three dimensional Riemannian manifold with nonnegative Ricci curvature and finite asymptotic curvature ratio.
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