Thesis (Ph.D.)--McMaster University (Canada), 1996. / Source: Dissertation Abstracts International, Volume: 58-06, Section: B, page: 3083. Adviser: McKenzie Wang.

<p> The aim of this thesis is to construct some smooth Einstein manifolds with nonzero Einstein constant, and then to investigate their topological and geometric properties.</p> <p> In the negative case, we are able to construct conformally compact Einstein metrics on 1. products of an arbitrary closed Einstein manifold and a certain even-dimensional ball bundle over products of Hodge Kähler-Einstein manifolds, 2. certain solid torus bundles over a single Fano Kähler-Einstein manifold. We compute the associated conformal invariants, i.e., the renormalized volume in even dimensions and the conformal anomaly in odd dimensions. As by-products, we obtain many Riemannian manifolds with vanishing Q-curvature.</p> <p> In the positive case, we are able to construct complete Einstein metrics on certain 3-sphere bundles over a Fano Kähler-Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base manifold is the complex projective plane.</p> / Thesis / Doctor of Philosophy (PhD)

We conduct a comprehensive investigative review of solution generating algorithms for the Einstein field equations governing the gravitational behaviour of an isolated neutral static spherical distribution of perfect fluid matter. Traditionally, the master field equation generated from the condition of pressure isotropy has been interpreted as a second order ordinary differential equation. However, since the pioneering work of Wyman (1949) it was observed that more success can be enjoyed by regarding the equation as a first order linear differential equation. There was a resurgence of the ideas of Wyman in 2000 and various researchers have been able to generate complete solutions to the field equations up to certain integrations. These have been accomplished by working in Schwarzschild (curvature) coordinates, isotropic coordinates, area coordinates and a coordinate system written in terms of the redshift parameter. We have utilised Durgapal–Banerjee (1983) coordinates and produced a new algorithm. The algorithm is used to generate new classes of perfect fluid solutions as well as to regain familiar particular solutions reported in the literature. We find that our solution is well behaved according to elementary physical requirements. The pressure vanishes for a certain radius and this establishes the boundary of the distribution. Additionally the pressure and energy density are both positive inside the radius. The energy conditions are shown to be satisfied and it is particularly pleasing to have the causality criterion satisfied to ensure that the speed of light is not exceeded by the speed of sound. We also report some new solutions using the algorithms proposed by Lake (2006). / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

The study of embeddings is applicable and signicant to higher dimensional theories of our universe, high-energy physics and classical general relativity. In this thesis we investigate local and global isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Theorems have been established that guarantee the existence of such embeddings. However, most known explicit results concern embedded spaces with relatively simple Ricci curvature. We consider the four-dimensional gravitational field of a global monopole, a simple non-vacuum space with a more complicated Ricci tensor, which is of theoretical interest in its own right, and occurs as a limit in Einstein-Gauss-Bonnet Kaluza-Klein black holes, and we obtain an exact solution for its embedding into Minkowski space. Our local embedding space can be used to construct global embedding spaces, including a globally at space and several types of cosmic strings. We present an analysis of the result and comment on its signicance in the context of induced matter theory and the Einstein-Gauss-Bonnet gravity scenario where it can be viewed as a local embedding into a Kaluza-Klein black hole. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate which embedded spaces admit bulks of a specific type. We show that the general Schwarzschild-de Sitter spacetime and the Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space with a particular metric form, and we discuss their five-dimensional solutions. Furthermore, we determine that the only spherically symmetric spacetime in retarded time coordinates that can be embedded into a particular Einstein bulk is the general Vaidya-de Sitter solution with constant mass. These analyses help to provide insight to the general embedding problem. We also consider the conformal Killing geometry of a five-dimensional Einstein space that embeds a static spherically symmetric spacetime, and we show how the Killing geometry of the embedded space is inherited by its bulk. The study of embedding properties such as these enables a deeper mathematical understanding of higher dimensional cosmological models and is also of physical interest as conformal symmetries encode conservation laws. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2012.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves [86]-88).

This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum. / Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus.

Orientador: Caio Jose Colletti Negreiros, Nir Cohen / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T00:38:44Z (GMT). No. of bitstreams: 1 Santos_EvandroCarlosFerreirados_D.pdf: 3261771 bytes, checksum: 1d6efb1b1f03e2549a501877f92a4952 (MD5) Previous issue date: 2005 / Resumo: O objetivo deste trabalho é contribuir para o estudo da geometria Hermitiana invariante das variedades bandeira. Estudamos a classe das métricas de Einstein sobre variedades bandeira. Neste trabalho apresentamos novas soluções para a equação de Einstein invariante sobre as variedades bandeira do tipo Az maximais e não-maximais. Considere W um subgrupo do grupo de WeyL Descrevemos uma ação natural de W sobre o conjunto das soluções da equação de Einstein invariante e provamos que esta ação deixa a equação e o conjunto solução invariantes. Determinamos a constante de Einstein de todas as métricas conhecidas e em alguns casos encontramos a métrica de Yamabe. Estudamos o funcional de Einstein- Hilbert e concluímos que toda métrica de Einstein invariante sobre uma variedade flag é estável. Usamos C- fibrações para provar que sobre JF(n), n > 4, uma métrica de Einstein (1,2)- simplética deve ser Kãhler. Fizemos uso da classificação das estruturas quase Hermitianas invariantes de San Martin- Negreiros e provamos que uma métrica de Einstein é Kãhler ou pertence à classe W1 EB W3. Isto implica em uma solução, no caso das variedades bandeira do tipo Az, para uma conjectura formulada por W. Ziller[17] / Abstract: The goal of this work is to contribute the study of invariant Hermitian geometry on flag manifolds. We study the class of Einstein metrics on flag manifolds. In this work we present new solutions for the invariant Einstein equation on flag manifolds, maximals or not, of Ai case. Let W a subgroup of the Weyl group. We described a natural action of W on the solution set of the Einstein equation, and we proved that W lefts the solution set invariant. We obtained the Einstein's constant of all the known metrics and in some cases we found the Yamabe metric. We studied the Einstein-Hilbert functional and we proved that all invariant Einstein metrics on a flag manifold are stable. Using C-fibrations we proved, in the case IF(n), n 2:: 4, if 9 is an invariant Einstein metric, and (1,2)-symplectic then 9 is Kãhler. According to San Martin-Negreiros's classification of all almost Hermitian structures on maximal flag manifolds we proved that an Einstein metric is Kãhler or belongs to W1 $ W3. This implies in a solution, in flag manifolds of Ai case, for a conjecture proposed by W. Ziller[17] / Doutorado / Geometria e Topologia / Doutor em Matemática

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-11-30T07:33:27Z No. of bitstreams: 2 Tese - Márcio Lemes de Sousa - 2015.pdf: 2626758 bytes, checksum: 1e9e1b9d216bad33d6b5919afa54a4e4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-11-30T07:35:41Z (GMT) No. of bitstreams: 2 Tese - Márcio Lemes de Sousa - 2015.pdf: 2626758 bytes, checksum: 1e9e1b9d216bad33d6b5919afa54a4e4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-11-30T07:35:41Z (GMT). No. of bitstreams: 2 Tese - Márcio Lemes de Sousa - 2015.pdf: 2626758 bytes, checksum: 1e9e1b9d216bad33d6b5919afa54a4e4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) 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 (n􀀀1)-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 (n􀀀1)-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 (n􀀀1) - dimensional, e a fibra F é Ricci -flat.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-06-15T19:51:42Z No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-06-28T12:21:16Z (GMT) No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Made available in DSpace on 2016-06-28T12:21:16Z (GMT). No. of bitstreams: 2 Dissertação - Elismar Dias Batista - 2016.pdf: 1518873 bytes, checksum: 8375db389a2056c5849ee168f5efa5ce (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2016-03-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based on the articles [26] and [27], where we studied Einstein manifolds and gradient Ricci soliton with twisted product structure. As a result, we prove the following: if M is an Einstein warped product space with nonpositive scalar curvature and compact base, then M is a Riemannian product space. Besides, we show that the Riemannian product Rp×F is a gradient Ricci soliton if and only if F is Ricci soliton gradient. Then, we show that the warped product R×f B is gradient Ricci solitons with f ′′ 6= 0, therefore F is Einstein. By using these results, we build nontrivial examples of gradient Ricci soliton where the fiber is either an Einstein manifold or a nontrivial gradient Ricci soliton. / Este trabalho está baseado nos artigos [26] e [27], onde estudamos Variedades de Einstein e gradiente Ricci solitons com estrutura de produto torcido. Provamos que: se M é um produto torcido Einstein com curvatura escalar não positiva e base compacta, então a função torção é constante, ou seja, o produto torcido é Riemanniano. Mostramos ainda que o produto Riemanniano Rp ×F é um gradiente Ricci soliton se e somente se F for gradiente Ricci soliton. Em seguida, mostramos que se o produto torcido R×f F for gradiente Ricci soliton com f ′′(t) 6= 0, então F é Einstein. Usando estes resultados construímos exemplos de gradiente Ricci soliton não trivial com a fibra sendo Einstein ou gradiente Ricci soliton não trivial. Finalmente consideramos o produto torcido Lorentziano sendo gradiente Ricci soliton e obtivemos critérios análogos ao Riemanniano para que F seja Einstein ou gradiente Ricci soliton.

Orientadores: Caio Jose Colleti Negreiros, Nir Cohen / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T14:44:13Z (GMT). No. of bitstreams: 1 Silva_NeitonPereirada_D.pdf: 4231710 bytes, checksum: af4dc57e0a7215547662f87d1744bb27 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho encontramos todas as métricas de Einstein invariantes em quatro famílias de variedades bandeira do tipo B1 e C1. Os nossos resultados são consistentes com a conjectura de Wang e Ziller sobre a finitude das métricas de Einstein. O nosso método para resolver as equações de Einstein e baseado nas simetrias do sistema algébrico. Obtemos os sistemas algébricos de Einstein para variedades bandeira generalizadas do tipo B1 C1e G2. Estes sistemas são as condições necessárias e suficientes para métricas invariantes nessas variedades serem Einstein. Os sistemas algébricos que obtivemos generalizam as equações de Einstein obtidas por Sakane nos casos maximais. As equações nos casos Al e Dl foram obtidas por Arvanitoyeorgos. Calculamos o conjunto das trazes para as variedades bandeira generalizadas dos grupos de Lie clássicos. Assim estendemos à essas variedades certos resultados sobre estruturas Hermitianas invariantes obtidos por San Martin, Cohen e Negreiros. / Abstract: In this work we and all the invariant Einstein metrics on four families of ag manifolds of type Bl and Cl. Our results are consistent with the finiteness conjecture of Einstein metrics proposed by Wang and Ziller. Our approach for solving the Einstein equations is based on the symmetries of the algebraic system. We obtain the Einstein algebraic systems for the generalized ag manifolds of type Bl, Cl and G2. These systems are necessary and sufficient conditions for invariant metrics on these manifolds to be Einstein. The algebraic systems that we obtained generalize the Einstein equations obtained by Sakane in the maximal cases. The equations in the cases Al and Dl were obtained by Arvanitoyeorgos. We calculate all the t-roots on the generalized ag manifolds of the classical Lie groups. Thus we extend to these manifolds certain results on invariant structures Hermitian obtained by San Martin, Cohen and Negreiros. / Doutorado / Geometria Diferencial / Doutor em Matemática