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Harst, Ulrich [Verfasser]. "Investigations on asymptotic safety of metric, tetrad and Einstein-Cartan gravity / Ulrich Harst." Mainz : Universitätsbibliothek Mainz, 2013. http://d-nb.info/1032940662/34.
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Champion, Daniel James. "Mobius Structures, Einstein Metrics, and Discrete Conformal Variations on Piecewise Flat Two and Three Dimensional Manifolds." Diss., The University of Arizona, 2011. http://hdl.handle.net/10150/145313.
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Spherical, Euclidean, and hyperbolic simplices can be characterized by the dihedral angles on their codimension-two faces. These characterizations analyze the Gram matrix, a matrix with entries given by cosines of dihedral angles. Hyperideal hyperbolic simplices are non-compact generalizations of hyperbolic simplices wherein the vertices lie outside hyperbolic space. We extend recent characterization results to include fully general hyperideal simplices. Our analysis utilizes the Gram matrix, however we use inversive distances instead of dihedral angles to accommodate fully general hyperideal simplices.For two-dimensional triangulations, an angle structure is an assignment of three face angles to each triangle. An angle structure permits a globally consistent scaling provided the faces can be simultaneously scaled so that any two contiguous faces assign the same length to their common edge. We show that a class of symmetric Euclidean angle structures permits globally consistent scalings. We develop a notion of virtual scaling to accommodate spherical and hyperbolic triangles of differing curvatures and show that a class of symmetric spherical and hyperbolic angle structures permit globally consistent virtual scalings.The double tetrahedron is a triangulation of the three-sphere obtained by gluing two congruent tetrahedra along their boundaries. The pentachoron is a triangulation of the three-sphere obtained from the boundary of the 4-simplex. As piecewise flat manifolds, the geometries of the double tetrahedron and pentachoron are determined by edge lengths that gives rise to a notion of a metric. We study notions of Einstein metrics on the double tetrahedron and pentachoron. Our analysis utilizes Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds.A notion of conformal structure on a two dimensional piecewise flat manifold is given by a set of edge constants wherein edge lengths are calculated from the edge constants and vertex based parameters. A conformal variation is a smooth one parameter family of the vertex parameters. The analysis of conformal variations often involves the study of degenerating triangles, where a face angle approaches zero. We show for a conformal variation that remains weighted Delaunay, if the conformal parameters are bounded then no triangle degenerations can occur.
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Wink, Matthias. "Ricci solitons and geometric analysis." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:3aae2c5e-58aa-42da-9a1b-ec15cacafdad.
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This thesis studies Ricci solitons of cohomogeneity one and uniform Poincaré inequalities for differentials on Riemann surfaces. In the two summands case, which assumes that the isotropy representation of the principal orbit consists of two inequivalent Ad-invariant irreducible summands, complete steady and expanding Ricci solitons have been detected numerically by Buzano-Dancer-Gallaugher-Wang. This work provides a rigorous construction thereof. A Lyapunov function is introduced to prove that the Ricci soliton metrics lie in a bounded region of an associated phase space. This also gives an alternative construction of non-compact Einstein metrics of non-positive scalar curvature due to Böhm. It is explained how the asymptotics of the Ricci flat trajectories induce Böhm's Einstein metrics on spheres and other low dimensional spaces. A numerical study suggests that all other Einstein metrics of positive scalar curvature which are induced by the generalised Hopf fibrations occur in an entirely non-linear regime of the Einstein equations. Extending the theory of cohomogeneity one steady and expanding Ricci solitons, an estimate which allows to prescribe the growth rate of the soliton potential at any given time is shown. As an application, continuous families of Ricci solitons on complex line bundles over products of Fano Kähler Einstein manifolds are constructed. This generalises work of Appleton and Stolarski. The method also applies to the Lü-Page-Pope set-up and allows to cover an optimal parameter range in the two summands case. The Ricci soliton equation on manifolds foliated by torus bundles over products of Fano Kähler Einstein manifolds is discussed. A rigidity theorem is obtained and a preserved curvature condition is discovered. The cohomogeneity one initial value problem is solved for m-quasi-Einstein metrics and complete metrics are described. Lp-Poincaré inequalities for k-differentials on closed Riemann surfaces are shown. The estimates are uniform in the sense that the Poincaré constant only depends on p ≥1, k ≥ 2 and the genus γ ≥ 2 of the surface but not on its complex structure. Examples show that the analogous estimate for 1-differentials cannot be uniform. This part is based on joint work with Melanie Rupflin.
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Welly, Adam. "The Geometry of quasi-Sasaki Manifolds." Thesis, University of Oregon, 2016. http://hdl.handle.net/1794/20466.
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Let (M,g) be a quasi-Sasaki manifold with Reeb vector field xi. Our goal is to understand the structure of M when g is an Einstein metric. Assuming that the S^1 action induced by xi is locally free or assuming a certain non-negativity condition on the transverse curvature, we prove some rigidity results on the structure of (M,g).
Naturally associated to a quasi-Sasaki metric g is a transverse Kahler metric g^T. The transverse Kahler-Ricci flow of g^T is the normalized Ricci flow of the transverse metric. Exploiting the transverse Kahler geometry of (M,g), we can extend results in Kahler-Ricci flow to our transverse version. In particular, we show that a deep and beautiful theorem due to Perleman has its counterpart in the quasi-Sasaki setting.
We also consider evolving a Sasaki metric g by Ricci flow. Unfortunately, if g(0) is Sasaki then g(t) is not Sasaki for t>0. However, in some instances g(t) is quasi-Sasaki. We examine this and give some qualitative results and examples in the special case that the initial metric is eta-Einstein.
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Fanaai, Hamidreza. "Flot géodésique, mesures invariantes et métriques d'Einstein." Université Joseph Fourier (Grenoble ; 1971-2015), 1997. http://www.theses.fr/1997GRE10278.
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Nous etudions le probleme de conjugaison des flots geodesiques dans deux cas differents. Dans le premier chapitre, nous considerons les varietes riemanniennes compactes de courbure sectionnelle strictement negative et dans le deuxieme chapitre nous traitons le cas des nilvarietes de rang deux. Nous etudions aussi a la fin du premier chapitre, le probleme de l'invariance par symetrie des mesures de patterson-sullivan et harmoniques reliees au flot geodesique. Le dernier chapitre de cette these est consacre a l'etude de varietes homogenes d'einstein de courbure scalaire negative ou nous donnons quelques exemples de telles varietes en etudiant les algebres de lie nilpotentes de rang deux.
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Edmonds, Bartlett Douglas Jr. "Approaching the Singularity in Gowdy Universes." VCU Scholars Compass, 2006. http://scholarscompass.vcu.edu/etd/1083.
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It has been shown that the cosmic censorship conjecture holds for polarized Gowdy spacetimes. In the more general, unpolarized case, however, the question remains open. It is known that cylindrically symmetric dust can collapse to form a naked singularity. Since Gowdy universes comprise gravitational waves that are locally cylindrically symmetric, perhaps these waves can collapse onto a symmetry axis and create a naked singularity. It is known that in the case of cylindrical symmetry, event horizons will not form under gravitational collapse, so the formation of a singularity on the symmetry axis would be a violation of the cosmic censorship conjecture.To search for cosmic censorship violation in Gowdy spacetimes, we must have a better understanding of their singularities. It is known that far from the symmetry axes, the spacetimes are asymptotically velocity term dominated, but this property is not known to hold near the axes. In this thesis, we take the first steps toward understanding on and near axis behavior of Gowdy spacetimes with space-sections that have the topology of the three-sphere. Null geodesic behavior on the symmetry axes is studied, and it is found that in some cases, a photon will wrap around the universe infinitely many times on its way back toward the initial singularity.
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Gaset, Rifà Jordi. "A multisymplectic approach to gravitational theories." Doctoral thesis, Universitat Politècnica de Catalunya, 2018. http://hdl.handle.net/10803/620740.
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The theories of gravity are one of the most important topics in theoretical physics and mathematical physics nowadays. The classical formulation of gravity uses the Hilbert-Einstein Lagrangian, which is a singular second-order Lagrangian; hence it requires a geometric theory for second-order field theories which leads to several difficulties. Another standard formulation is the Einstein-Palatini or Metric-Affine, which uses a singular first order Lagrangian.
Much work has been done with the aim of establishing the suitable geometrical structures for describing classical field theories. In particular, the multisymplectic formulation is the most general of all of them and, in recent years, some works have considered a multisymplectic approach to gravity. This formulation allows us to study and better understand several inherent characteristics of the models of gravity.
The aim of this thesis is to use the multisymplectic formulation for first and second-order field theories in order to obtain a complete covariant description of the Lagrangian and Hamiltonian formalisms for the Einstein-Hilbert and the Metric-Affine models, and explain their characteristics; in particular: order reduction, constraints, symmetries and gauge freedom.
Some properties of multisymplectic field theories have been developed in order to study the models. We have established the constraints generated by the projectability of the Poincaré-Cartan form. These constraints are related to the fact that the higher order velocities are strong gauge vector fields. The concept of gauge freedom for field theories also has been analyzed. We propose to use the term "gauge'' to refer to the non-regularity of the Poincaré-Cartan form. Therefore, the multiple solutions are characterized by two sources: the gauge related one, arising from gauge symmetries and related to the non-regularity; and the non-gauge related one, which arises exclusively from field theories.
We studied in detail two models of gravity: the Einstein-Hilbert model and the Metric-Affine (or Einstein-Palatini) model. In both cases, a covariant Hamiltonian multisymplectic formalism has been presented. In every situation, we find the final submanifold where solutions exist, and we explicitly write all semi-holonomic multivector fields solution of the field equations. The natural Lagrangian symmetries are presented aswell. Furthermore, we emphasize different aspects in each model:
The Einstein-Hilbert model is a singular second order field theory which, as a consequence of its non-regularity, it is equivalent to a regular first order theory. For this model we have presented the unified Lagrangian-Hamiltonian formalism. We have also considered the presence of energy-matter sources and we show how some relevant geometrical and physical characteristics of the theory depend on the source's type.
The Metric-Affine model is a singular first order field theory which has a gauge symmetry. We recover and study this gauge symmetry, showing that there are no more. The constraints of the system are presented and analysed. Using the gauge freedom and the constraints, we establish the geometric relation between the Einstein-Palatini and the Einstein-Hilbert models, including the relation between the holonomic solutions in both formalisms. We also present a Hamiltonian model involving only the connection which is equivalent to the Hamiltonian Metric-Affine formalism.Les teories de la gravetat són un dels temes més importants en física teòrica i física matemàtica avui en dia. La formulació clàssica de la gravetat utilitza el Lagrangià de Hilbert-Einstein, el qual és un Lagrangià singular de segon ordre; per tant requereix una teoria geomètrica per teories de camp de segon ordre, que comporten diverses dificultats. Una altra formulació estàndard és la d'Einstein-Palatini o Mètrica-Afí, la qual utilitza un Lagrangià singular de primer ordre. S'ha treballat molt per establir les estructures geomètriques adients per descriure teories de camps clàssiques. Particularment, la formulació multisimplèctica és la més general de totes i, recentment alguns treballs han considerat la gravetat des de un punt de vista multisimplèctic. Aquesta formulació ens permet estudiar i entendre millor diverses característiques inherents dels models gravitatoris. L'objectiu d'aquesta tesi és utilitzar la formulació multisimplèctica per a teories de camps de primer i segon ordre per obtenir una descripció covariant completa dels formalismes Lagrangià i Hamiltonià per als models d'Einstein-Hilbert i Mètrica-Afí, i explicar les seves característiques. Concretament: reducció de l'ordre, restriccions, simetries i llibertat gauge. Algunes propietats de les teories de camps multisimplèctiques han estat desenvolupades per estudiar els models. S'han establert les restriccions generades per la projectabilitat de la forma de Poincaré-Cartan. Aquestes restriccions tenen relació amb el fet que les velocitats d'ordre superior són camps vectorials gauge forts. El concepte de llibertat gauge per a teories de camps també ha estat analitzat. Es proposa la utilització del terme "gauge" per fer referència a la no regularitat de les formes de Poincaré-Cartan. Per tant, les múltiples solucions es caracteritzen a partir de dues fonts: la relativa al gauge, que està relacionada amb la no regularitat, i altres fonts no relacionades amb el gauge que són exclusives de teories de camps. S'ha estudiat en detall dos models de gravetat: el model d'Einstein-Hilbert i el de Mètrica-Afí (o Einstein-Palatinti). En ambdós casos s'ha presentat una formulació covariant multisimplèctica Hamiltoniana. En tots els casos trobem la subvarietat final on les solucions existeixen, i escrivim explícitament tots els camps multivectorials sem-holònoms solució de les equacions de camp. També presentem les simetries Lagrangianes naturals. A més emfatitzem aspectes diferents en cada model: El model d'Einstein-Hilbert és una teoria de camp singular de segon ordre, la qual, com a conseqüència de la seva no regularitat, és equivalent a una teoria regular de primer ordre. Per aquest model hem presentat el formalisme unificat Lagrangià-Hamiltonià. També hem considerat la presència de fonts d'energia-matèria i es mostra com algunes característiques físiques i geomètriques rellevants de la teoria depenen del tipus de font. El model Mètrica-Afí és una teoria de camps singular de primer ordre que té una simetria gauge. Es recupera i s'estudia aquesta simetria gauge mostrant que és única. Les lligadures del sistema són presentades i analitzades. Utilitzant la llibertat gauge i les lligadures, s'estableix la relació geomètrica entre els models d'Einstein-Palatini i d'Einstein-Hilbert, inclosa la relació entre les solucions holònomes en ambdós formalismes. També es presenta un model Hamiltonià, que conté únicament la connexió, equivalent al formalisme Mètrica-Afí Hamiltonià
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Roth, John Charles. "Perturbations of Kähler-Einstein metrics /." Thesis, Connect to this title online; UW restricted, 1999. http://hdl.handle.net/1773/5737.
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Desa, Zul Kepli Bin Mohd. "Riemannian manifolds with Einstein-like metrics." Thesis, Durham University, 1985. http://etheses.dur.ac.uk/7571/.
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In this thesis, we investigate properties of manifolds with Riemannian metrics which satisfy conditions more general than those of Einstein metrics, including the latter as special cases. The Einstein condition is well known for being the Euler- Lagrange equation of a variational problem. There is not a great deal of difference between such metrics and metrics with Ricci tensor parallel for the latter are locally Riemannian products of the former. More general classes of metrics considered include Ricci- Codazzi and Ricci cyclic parallel. Both of these are of constant scalar curvature. Our study is divided into three parts. We begin with certain metrics in 4-dimensions and conclude our results with three theorems, the first of which is equivalent to a result of Kasner [Kal] while the second and part of the third is known to Derdzinski [Del.2].Next we construct the metrics mentioned above on spheres of odd dimension. The construction is similar to Jensen's [Jel] but more direct and is due essentially to Gray and Vanhecke [GV]. In this way we obtain .beside the standard metric, the second Einstein metric of Jensen. As for the Ricci- Codazzi metrics, they are essentially Einstein, but the Ricci cyclic parallel metrics seem to form a larger class. Finally, we consider subalgebras of the exceptional Lie algebra g2. Making use of computer programmes in 'reduce' we compute all the corresponding metrics on the quotient spaces associated with G2.
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Pedersen, H. "Geometry and magnetic monopoles : Constructions of Einstein metrics and Einstein-Weyl geometries." Thesis, University of Oxford, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.353118.
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Li, Long. "On the Uniqueness of singular Kahler-Einstein metrics." Thesis, State University of New York at Stony Brook, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3632422.
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Bando and Mabuchi proved the uniqueness of Kaehler-Einstein metrics on Fano manifolds up to a holomorphic automorphism in 1987. Then recently Berndtsson generalized the uniqueness result of Kaehler-Einstein metrics to bounded potentials. We give a new proof of the Bando-Mabuch-Berndtsson uniqueness theorem in a different aspect, based on a new technique developed from Chen's C1,1 geodesic and Futaki's spectral formula. Finally, the uniqueness of the conical Kaehler-Einstein metrics will be discussed under the assumption of properness of twisted Ding-functional.
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Santos, Evandro Carlos Ferreira dos. "Metricas de Einstein em variedades bandeira." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306786.
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Orientador: Caio Jose Colletti Negreiros, Nir CohenTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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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
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Kerr, Gary Drummond. "Algebraically special Einstein spaces : Kerr-Schild metrics and homotheties." Thesis, Queen Mary, University of London, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.300621.
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Araujo, Fatima. "Einstein homogeneous Riemannian fibrations." Thesis, University of Edinburgh, 2008. http://hdl.handle.net/1842/4375.
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This thesis is dedicated to the study of the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for the existence of Einstein metrics with totally geodesic fibers in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We investigate the existence of such Einstein metrics for invariant bisymmetric fibrations of maximal rank, i.e., when both the base and the fiber are symmetric spaces and the base is an isotropy irreducible space of maximal rank. We find this way new Einstein metrics. For such spaces we describe explicitly the isotropy representation in terms subsets of roots and compute the eigenvalues of the Casimir operators of the fiber along the horizontal direction. Results for compact simply connected 4-symmetric spaces of maximal rank follow from this. Also, new invariant Einstein metrics are found on Kowalski n-symmetric spaces.
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Borges, Laena Furtado. "Sobre rigidez de métricas quasi-Einstein." Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/6965.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this work, we will present some concepts of quasi-Einstein metrics. From this, we will enunciate and demonstrate rigidity results for quasi-Einstein metrics until we have enough material to demonstrate a stiffness result for quasi-Einstein metrics of dimension two. Finally, we will give some concepts of Kähler metrics, prove a theorem and finally demonstrate a corollary that connects the main theorem of our work with Kähler metrics.
Nesse trabalho, apresentaremos alguns conceitos de métricas quasi-Einstein. A partir disso, enunciaremos e demonstraremos resultados de rigidez para métricas quasi-Einstein, até que tenhamos material suficiente para a demonstração de um resultado de rigidez para métricas quasi-Einstein em dimensão dois. Por fim, daremos alguns conceitos de métricas kähler, provaremos um teorema e por fim demonstraremos um corolário que conecta o teorema principal do nosso trabalho com as métricas Kähler.
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Akbaba, Esin. "Einstein Aether Gravity." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/2/12610898/index.pdf.
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In this thesis, we review some basic properties of the Einstein-aether gravity. We derive
the field equations from an action and study a subclass of this theory corresponding to
the Einstein-Maxwell like theory. We also show that the Gödel type metrics are also exact solutions of this theory. Furthermore, we determine the observational constraints on the dimensionless preferred parameters of this theory using the parametrized post- Newtonian formalism. We stress that none of calculations and discussions are original in this thesis.
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Wu, Damin Ph D. Massachusetts Institute of Technology. "Higher canonical asymptotics of Kähler-Einstein metrics on quasi-projective manifolds." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/33600.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 61-64).
In this thesis, we derive the asymptotic expansion of the Kiihler-Einstein metrics on certain quasi-projective varieties, which can be compactified by adding a divisor with simple normal crossings. The weighted Cheng-Yau Hilder spaces and the log-filtrations based on the bounded geometry are introduced to characterize the asymptotics. We first develop the analysis of the Monge-Ampere operators on these weighted spaces. We construct a family of linear elliptic operators which can be viewed as certain conjugacies of the specially linearized Monge-Ampbre operators. We derive a theorem of Fredholm alternative for such elliptic operators by the Schauder theory and Yau's generalized maximum principle. Together these results derive the isomorphism theorems of the Monge-Ampbre operators, which imply that the Monge-Ampere operators preserve the log-filtration of the Cheng-Yau Holder ring. Next, by choosing a canonical metric on the submanifold, we construct an initial Kidhler metric on the quasi-projective manifold such that the unique solution of the Monge-Ampere equation belongs to the weighted -1 Cheng-Yau Hölder ring. Moreover, we generalize the Fefferman's operator to act on the volume forms and obtain an iteration formula.
(cont.) Finally, with the aid of the isomorphism theorems and the iteration formula we derive the desired asymptotics from the initial metric. Furthermore, we prove that the obtained asymptotics is canonical in the sense that it is independent of the extensions of the canonical metric on the submanifold.
by Damin Wu.
Ph.D.
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Silva, Neiton Pereira da. "Metricas de Einstein e estruturas Hermitianas invariantes em variedades bandeira." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306785.
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Orientadores: Caio Jose Colleti Negreiros, Nir CohenTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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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
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Junior, Ernani de Sousa Ribeiro. "A geometria das mÃtricas tipo-Einstein." Universidade Federal do CearÃ, 2011. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=6655.
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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel SuperiorConselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
O objetivo deste trabalho à estudar a geometria das mÃtricas tipo-Einstein (solitons de Ricci, quase solitons de Ricci e mÃtricas quasi-Einstein). Mais especificamente, vamos obter equaÃÃes de estrutura, exemplos, fÃrmulas integrais e estimativas que permitirÃo caracterizar estas classes de mÃtricas.
The purpose of this work is study the geometric of the like-Einstein metrics (Ricci soliton, almost Ricci solitons and quasi-Einstein metrics). More specifically, we obtain structure equations, examples, integral formulae and estimates that will enable characterize these classes of metrics.
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Griffiths, Hugh Norman. "Self-dual metrics on toric 4-manifolds : extending the Joyce construction." Thesis, University of Edinburgh, 2009. http://hdl.handle.net/1842/3969.
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Toric geometry studies manifolds M2n acted on effectively by a torus of half their dimension, Tn. Joyce shows that for such a 4-manifold sufficient conditions for a conformal class of metrics on the free part of the action to be self-dual can be given by a pair of linear ODEs and gives criteria for a metric in this class to extend to the degenerate orbits. Joyce and Calderbank-Pedersen use this result to find representatives which are scalar flat K¨ahler and self-dual Einstein respectively. We review some results concerning the topology of toric manifolds and the construction of Joyce metrics. We then extend this construction to give explicit complete scalar-flat K¨ahler and self-dual Einstein metrics on manifolds of infinite topological type, and to find a new family of Joyce metrics on open submanifolds of toric spaces. We then give two applications of these extensions — first, to give a large family of scalar flat K¨ahler perturbations of the Ooguri-Vafa metric, and second to search for a toric scalar flat K¨ahler metric on a neighbourhood of the origin in C2 whose restriction to an annulus on the degenerate hyperboloid {(z1, z2)|z1z2 = 0} is the cusp metric.
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Piolanti, Simone. "Il formalismo ADM per la metrica FLRW." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2020.
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Il tema centrale di questo elaborato è la cosmologia, affrontata in ambito di relatività generale applicando il formalismo ADM al modello FLRW. Il problema consiste nella risoluzione delle equazioni di Einstein in presenza di materia. Sfruttando quindi il formalismo ADM sono innanzitutto definite le formulazioni lagrangiana ed hamiltoniana di una teoria in ambito di relatività generale. In particolare sono descritte tali formulazioni per le equazioni di Einstein. Segue come applicazione l'esempio della particella libera relativistica interpretata come teoria di campo in una dimensione. È infine trattato il caso di interesse: si considera la metrica FLRW per universo piatto e sono risolte le equazioni di Einstein calcolate a partire da un'azione in cui la materia è descritta da un campo scalare senza massa di minimo accoppiamento.
Il risultato ottenuto descrive due possibili universi: uno in espansione e uno in contrazione. In particolare, l'evoluzione del fattore di scala rispetto al tempo proprio del sistema è descritto dalla relazione α^3 (τ)=α^3 (0)(1±τ/τ_c).
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Martinez, Garcia Jesus. "Dynamic alpha-invariants of del Pezzo surfaces with boundary." Thesis, University of Edinburgh, 2013. http://hdl.handle.net/1842/8090.
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The global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an important role when studying the geometry of Fano varieties. In particular, Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. First we extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical. In particular, we give a very explicit classifiation of very singular anticanonical pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree, where the boundary is any smooth elliptic curve C. Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities of angle β along C.
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Delcroix, Thibaut. "Métriques de Kähler-Einstein sur les compactifications de groupes." Thesis, Université Grenoble Alpes (ComUE), 2015. http://www.theses.fr/2015GREAM046/document.
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Le résultat principal de cette thèse est l'obtention d'une condition nécessaire et suffisante pour l'existence d'une métrique de Kähler-Einstein sur une compactification bi-équivariante lisse et Fano d'un groupe complexe réductif connexe. Ces variétés comprennent les variétés toriques et les compactifications magnifiques de groupes semisimples adjoints.Dans la première partie de ce travail sont développés les outils nécessaires à l'étude de l'existence de métriques de Kähler-Einstein sur ces variétés. Nous calculons en particulier la Hessienne complexe d'une fonction $Ktimes K$-invariante sur la complexification d'un groupe compact $K$. Nous associonségalement, à toute métrique invariante à courbure positive sur un fibré linéarisé ample sur une compactification de groupe, une fonction convexe dont le comportement asymptotique est prescrit. Ceci est utilisé une première fois pour obtenir une formule pour l'invariant alpha d'un fibré en droite ample sur une compactification de groupe Fano. Cette formule est obtenue par le calcul des seuils log canoniques des métriques hermitiennes invariantes à courbure positive, et induit, dans le cas particulier des variétés toriques, un résultat obtenu auparavant, figurant dans l'article par ailleurs inclus en appendice de la thèse.Nous prouvons ensuite le résultat principal en obtenant des estimées $C^0$ le long de la méthode de continuité, en se ramenant à une équation de Monge-Ampèreréelle sur un cône. La condition obtenue est que le barycentre du polytope associé à la compactification de groupe, par rapport à la mesure de Duistermaat-Heckman, doit être dans une zone particulière du polytope. Cette condition peut être vérifiée sur les exemples, donne de nouveaux exemples de variétés deKähler-Einstein Fano, et donne aussi un exemple qui n'admet aucun soliton de Kähler-Ricci. Nous calculons de plus la plus grande borne inférieure de Ricci lorsqu'il n'y a pas de métrique de Kähler-EinsteinThe main result of this work is a necessary and sufficient condition for the existence of a Kähler-Einstein metric on a smooth and Fano bi-equivariant compactification of a complex connected reductive group. Examples of such varieties include wonderful compactifications of adjoint semisimple groups.The tools needed to study the existence of Kähler-Einstein metrics on these varieties are developed in the first part of the work, including a computation of the complex Hessian of a $Ktimes K$-invariant function on the complexification of a compact group $K$. Another step is to associate to any non-negatively curved invariant hermitian metric on an ample linearized line bundle on a group compactification a convex function with prescribed asymptotic behavior. This is used a first time to derive a formula for the alpha invariantof an ample line bundle on a Fano group compactification. This formula is obtained through the computation of the log canonical thresholds of any non-negatively curved invariant hermitian metric, and gives the sameresult, for toric manifolds, as the one we obtained before, in an article that is included in this thesis as an appendix.Then we prove the main result by obtaining $C^0$ estimates along the continuity method, using the tools developed to reduce to a real Monge-Ampère equation on a cone. The condition obtained is that the barycenter of the polytope associated to the group compactification, with respect to the Duistermaat-Heckman measure, lies in a certain zone in the polytope. This condition can be checked on examples, gives new examples of Fano Kähler-Einstein manifolds, and also gives an example that admits no Kähler-Ricci solitons. We also compute the greatest Ricci lower bound when there are no Kähler-Einstein metrics
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Filho, JoÃo Francisco da Silva. "Solitons de Ricci e mÃtricas quasi-Einstein em variedades homogÃneas." Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11123.
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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgicoCoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
Este trabalho tem como objetivo principal estudar os solitons de Ricci e as mÃtricas quasi-Einstein em variedades riemannianas homogÃneas e simplesmente conexas, enfatizando problemas em dimensÃes trÃs e quatro, procurando caracterizar e descrever explicitamente tais estruturas, obtendo resultados de existÃncia, unicidade e consequentemente, construir novos exemplos sobre essas classes de variedades. A descriÃÃo mencionada, consiste basicamente em determinar condiÃÃes que garantam existÃncia e explicitar a famÃlia de campos de vetores que geram todas essas possÃveis estruturas, relacionando-os entre si e identificando quais desses campos de vetores sÃo do tipo gradiente. Devemos ressaltar que a parte do trabalho que corresponde Ãs variedades homogÃneas de dimensÃo trÃs considera a classificaÃÃo relativa à dimensÃo do grupo de isometrias, enquanto a parte que corresponde Ãs variedades homogÃneas de dimensÃo quatro, contempla apenas uma subclasse das variedades homogÃneas de dimensÃo quatro que à constituÃda pelas variedades solÃveis tipo-Lie, ou seja, grupos de Lie solÃveis, simplesmente conexos e munidos de mÃtrica invariante à esquerda.
The purpose of this work is study Ricci solitions and quasi-Einstein metrics on simply connected homogeneous Riemannian manifolds, with emphasis in problems in three and four dimensions, trying to characterize and to describe explicitly such structures, getting results of existence, uniqueness and consequently, build new examples on these class of manifolds. The quoted description consists basically in to obtain conditions that ensure the existence and show explicitly the family of vector fields that generate each of these structures, relating them identifying what of these vector fields are gradient. We should highlight that in the part of this work that corresponds to homogeneous three manifolds, we will consider the classification relative to dimension of isometry group, while in the part that corresponds to homogeneous four manifolds, we treat only the solvable geometry Lie type, namely, the simply connected solvable Lie group with left invariants metrics.
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Carlos, Elaine Sampaio de Sousa. "A geometria dos sÃlitons de Ricci compactos." Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11129.
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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel SuperiorConselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
O objetivo deste trabalho à estudar a geometria dos sÃlitons de Ricci compactos, os quais correspondem as soluÃÃes auto-similires do fluxo de Ricci. AlÃm disso, essas variedades podem ser vistas como uma generalizaÃÃo das mÃtricas de Einstein. Neste trabalho, mostraremos que todo sÃliton de Ricci compacto tem curvatura escalar positiva. Alem disso, mostraremos que o seu grupo fundamental à sempre finito. Em particular, apresentaremos uma prova feita por Perelman [19] que todo sÃliton de Ricci compacto à do tipo gradiente
The aim of this work is to study the geometry of the compact Ricci soliton, which correspond to self-similar solution of the Ricci flow. These manifolds are natural generalization to Einstein metrics. Here we shall prove that every compact Ricci soliton has positive scalar curvature. Moreover, we show that its fundamental group is finite. Finally, we prove that every compact Ricci soliton must be gradient.
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Silva, Carlos Antonio Freitas da. "Construção explícita de métricas de Einstein-Finsler com curvatura flag não constante." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4520.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this dissertation we will study Finsler Geometry. In particular, we will study Randers Geometry that which can be viewed as Riemannian Geometry with a pertubation. Furthermore Randers metrics are also obtained as solution to Zermelo’s Navigation Problem. We will also use classification theorems of Randers metrics of constant flag curvature and Einstein Randers metrics in terms of Zermelo’s Navigation Problem. Using Randers metrics we are going to construct a 3-parameter family of Einstein-Finsler metrics with non-constant flag curvature and to get such family we use a Killing vector field and a Riemannian metric which is the Hawking Taub-NUT metric.
Neste trabalho estudaremos a Geometria de Finsler. Em particular, estudaremos a Geometria de Randers que pode ser visto como a mais simples perturbação da Geometria Riemanniana. Além disso, veremos também que métricas de Randers podem ser obtidas como soluções do Problema Navegacional de Zermelo. Utilizaremos também resultados que caracterizam métricas de Randers com curvatura flag constante e métricas de Randers do tipo Einstein em termos do Problema Navegacional de Zermelo. Usando métricas de Randers vamos construir uma família a 3 parâmetros de métricas de Einstein-Finsler com curvatura flag não constante e para obter tal família utilizaremos um campo de Killing e uma métrica Riemanniana que é a métrica de Hawking Taub-NUT.
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Bezerra, Kelton Silva. "Rigidity and unstability of hypersurfaces and an unicity theorem on semi-Rieamannian manifolds." Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=16530.
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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel SuperiorOur aim in this work is threefold. First, we get an extension, to the spherical case, of a theorem due to J. Simons, which concerns unstability of minimal cones constructed over a certain class of minimal submanifolds of the Euclidean sphere. Second, we classify the quasi-Einstein structures of the Riemannian product Hn x R. Third, we get a rigidity theorem for complete hypersurfaces into the De Sitter space, under certain conditions on the mean and scalar curvatures.
Este trabalho aborda trÃs problemas em Geometria Diferencial. Primeiro, obtemos uma extensÃo, para o caso esfÃrico, de um teorema devido a J. Simons sobre instabilidade de cones mÃnimos construÃdos sobre uma certa classe de subvariedades mÃnimas da esfera Euclidiana. Depois, classificamos as estruturas quasi-Einstein existentes sobre o produto Riemanniano Hn X R. Por fim, obtemos um teorema de rigidez para hipersuperfÃcies tipo-espaÃo completas do espaÃo de De Sitter, sob certas condiÃÃes sobre as curvaturas mÃdia e escalar, alÃm de uma condiÃÃo de integrabilidade.
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Silva, Antonio Kelson Vieira da. "Estimativas gradiente para autofunções do V-Laplaciano e métricas m-quasi-Einstein generalizadas compactas com bordo." reponame:Repositório Institucional da UFC, 2015. http://www.repositorio.ufc.br/handle/riufc/22559.
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SILVA, A. K. V. Estimativas gradiente para autofunções do V-Laplaciano e métricas m-quasi-Einstein generalizadas compactas com bordo. 2017. 40 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-18T14:49:03Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322426 bytes, checksum: d6abc5541598409191f635ad25e1f501 (MD5)
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The main of this work was to study properties of Riemannian when subjected to conditions on Bakry-Émery-Ricci tensor. Essentially we study two cases. In the first case, motivated by the work of Barros and Ribeiro Jr. (2014), He, Petersen and Wylie (2012) and Miao and Tam (2011), was introduced generalized m-quasi-Einstein metrics compact with boundary, where we get a result that classify these metrics; more specifically, assuming that gradient field of the exponential of potential function is a conformal vector field, we obtain that this must be a geodesic ball in a simply connected space form. That we get some results that implies when these are trivial metrics. In the second case, we work the Bakry-Émery-Ricci tensor bounded bellow, initially in a compact Riemannian, with or without boundary, and later on balls in complete Riemannian. With this study, we obtain gradient estimates for eigenfunctions of V-Laplacian operator, that generalize results of (Li, 2005) and (Li, 2015). Finally, as consequence theses results, we show an Harnack’s inequality.
Este trabalho tem como principal objetivo estudar propriedades de variedades Riemannianas quando submetidas a condições sobre tensores de Ricci-Bakry-Émery. Essencialmente estudamos dois casos. No primeiro caso, motivados pelos trabalhos de Barros e Ribeiro Jr (2014), He, Petersen e Wylie (2012) e por Miao e Tam (2011), introduzimos métricas m-quasi-Einstein generalizadas compactas com bordo, donde obtemos um resultado que garante uma classificação para estas métricas; mais precisamente, assumindo que o gradiente da exponencial da função potencial é um campo conforme, obtemos que aquela deve ser uma bola geodésica de uma forma espacial simplesmente conexa. Disso, obtemos alguns resultados em que garantimos quando estas métricas são triviais. No segundo caso, trabalhos o tensor de Ricci-Bakry-Émery limitado por baixo, inicialmente, em variedades Riemannianas compactas, com bordo ou sem bordo, e posteriormente, sobre bolas em variedades Riemannianas completas. Com esse estudo, obtivemos estimativas do gradiente para autofunções do operador V-Laplaciano, generalizando resultados de (Li, 2005) e (Li, 2015). Finalmente, como consequências desses resultados, exibimos uma desigualdade de Harnack.
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Montcouquiol, Grégoire. "Déformations de métriques Einstein sur des variétés à singularités coniques." Toulouse 3, 2005. http://www.theses.fr/2005TOU30205.
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Partant d'une cône-variété hyperbolique compacte de dimension n>2, on étudie les déformations de la métrique dans le but d'obtenir des cônes-variétés Einstein. Dans le cas où le lieu singulier est une sous-variété fermée de codimension 2 et que tous les angles coniques sont plus petits que 2pi, on montre qu'il n'existe pas de déformations Einstein infinitésimales non triviales préservant les angles coniques. Ce résultat peut s'interpréter comme une généralisation en dimension supérieure du célèbre théorème de Hodgson et Kerckhoff sur les déformations des cônes-variétés hyperboliques de dimension 3. Si tous les angles coniques sont inférieurs à pi, on donne ensuite une construction qui à chaque variation donnée des angles associe une déformation Einstein infinitésimale correspondanteStarting with a compact hyperbolic cone-manifold of dimension n>2, we study the deformations of the metric in order to get Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2pi, we show that there is no non-trivial infinitesimal Einstein deformations preserving the cone angles. This result can be interpreted as a higher-dimensional case of the celebrated Hodgson and Kerckhoff's theorem on deformations of hyperbolic 3-cone-manifolds. If all cone angles are smaller than pi, we also give a construction which associates to any variation of the angles a corresponding infinitesimal Einstein deformation
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Evangelista, Israel de Sousa. "Compact almost Ricci soliton, critical metrics of the total scalar curvature functional and p-fundamental tone estimates." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/23920.
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EVANGELISTA, I. S. Compact almost Ricci soliton, critical metrics of the total scalar curvature functional and p-fundamental tone estimates. 2017. 75 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-07-10T12:41:32Z No. of bitstreams: 1 2017_tese_isevangelista.pdf: 618771 bytes, checksum: 7e4bb8d9fd8825ef347e309171075037 (MD5)
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The present thesis is divided in three different parts. The aim of the first part is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive. The aim of the second part is to study the critical metrics of the total scalar curvature funcional on compact manifolds with constant scalar curvature and unit volume, for simplicity, CPE metrics. It has been conjectured that every CPE metric must be Einstein. We prove that the Conjecture is true for CPE metrics under a suitable integral condition and we also prove that it suffices the metric to be conformal to an Einstein metric. In the third part we estimate the p-fundamental tone of submanifolds in a Cartan-Hadamard manifold. First we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension one C 2-foliations of open subsets Ω of Riemannian manifolds M and obtain lower bounds estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of Ω.
A presente tese está dividida em três partes diferentes. O objetivo da primeira parte é provar que um quase soliton de Ricci compacto com tensor de Cotton nulo é isométrico a uma esfera canônica desde que uma das seguintes condições associadas ao tensor de Schouten seja válida: a segunda função simétrica é constante e positiva; duas funções simétricas consecutivas são múltiplas, não nulas, ou alguma função simétrica é constante e o tensor de Schouten é positivo. O objetivo da segunda parte é estudar as métricas críticas do funcional curvatura escalar total em variedades compactas com curvatura escalar constante e volume unitário, por simplicidade, métricas CPE. Foi conjecturado que toda métrica CPE deve ser Einstein. Prova-se que a conjectura é verdadeira para as métricas CPE sob uma condição integral adequada e também se prova que é suficiente que a métrica seja conforme a uma métrica Einstein. Na terceira parte, estima-se o p-tom fundamental de subvariedades em uma variedade tipo Cartan-Hadamard. Primeiramente, obtém-se estimativas por baixo para o p-tom fundamental de bolas geodésicas e em subvariedades com curvatura média limitada. Além disso, obtém-se estimativas do p-tom fundamental de subvariedades mínimas com certas condições sobre a norma da segunda forma fundamental. Por fim, estudam-se folheações de classe C 2 transversalmente orientadas de codimensão 1 de subconjuntos abertos Ω de variedades riemannianas M e obtêm-se estimativas por baixo para o ínfimo da curvatura média das folhas em termos do p-tom fundamental de Ω.
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Silva, Adam Oliveira da. "Rigidez de métricas críticas para funcionais riemannianos." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25969.
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SILVA, Adam Oliveira da. Rigidez de métricas críticas para funcionais riemannianos. 2017. 78 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-09-19T19:08:04Z No. of bitstreams: 1 2017_tese_aosilva.pdf: 481005 bytes, checksum: 2bdfc6ab68b042a5cfd4f67caf1e21e4 (MD5)
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The aim of this work is to study metrics that are critical points for some Riemannian functionals. In the first part, we investigate critical metrics for functionals which are quadratic in the curvature on closed Riemannian manifolds. It is known that space form metrics are critical points for these functionals, denoted by F t,s (g). Moreover, when s = 0, always Einstein metrics are critical to F t (g). We proved that under some conditions the converse is true. For instance, among others results, we prove that if n ≥ 5 and g is a Bach-flat critical metric to F −n/4(n−1) , with second elementary symmetric function of the Schouten tensor σ 2 (A) > 0, then g should be Einstein. Furthermore, we show that a locally conformally flat critical metric with some additional conditions are space form metrics. In the second part, we study the critical metrics to volume functional on compact Riemannian manifolds with connected smooth boundary. We call such critical points of Miao-Tam critical metrics due to the variational study making by Miao and Tam (2009). In this work, we show that the geodesics balls in space forms Rn , Sn and Hn have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that the boundary be an Einstein manifold. In the same spirit, we also extend a rigidity theorem due to Boucher et al. (1984) and Shen (1997) to n-dimensional static metrics with positive constant scalar curvature, which give us another way to get a partial answer to the Cosmic no-hair conjecture already obtained by Chrusciel (2003).
Este trabalho tem como principal objetivo estudar métricas que são pontos críticos de alguns funcionais Riemannianos. Na primeira parte, investigaremos métricas críticas de funcionais que são quadráticos na curvatura sobre variedades Riemannianas fechadas. É de conhecimento que métricas tipo formas espaciais são pontos críticos para tais funcionais, denotados aqui por F t,s (g). Além disso, no caso s = 0, métricas de Einstein são sempre críticas para F t (g). Provamos que sob algumas condições, a recíproca destes fatos são verdadeiras. Por exemplo, dentre outros resultados, provamos que se n ≥ 5 e g é uma métrica Bach-flat crìtica para F−n/4(n−1) com segunda função simétrica elementar do tensor de Schouten σ 2 (A) > 0, então g tem que ser métrica de Einstein. Ademais, mostramos que uma métrica crítica localmente conformemente plana, com algumas hipóteses adicionais, tem que ser tipo forma espacial. Na segunda parte, estudamos as métricas críticas do funcional volume sobre variedades Riemannianas compactas com bordo suave conexo. Chamamos tais pontos críticos de métricas críticas de Miao-Tam, devido ao estudo variacional feito por Miao e Tam (2009). Neste trabalho provamos que as bolas geodésicas das formas espaciais Rn , S n e H n possuem o valor máximo para o volume do bordo dentre todas as métricas críticas de Miao-Tam com bordo conexo, desde que o bordo seja uma variedade de Einstein. No mesmo sentido, também estendemos um teorema de rigidez devido à Boucher et al. (1984) e Shen (1997) para métricas estáticas de dimensão n e com curvatura escalar constante positiva, o qual nos fornece outra maneira para obter uma resposta parcial para a Cosmic no-hair conjecture já obtida por Chrusciel (2003).
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Delgove, François. "Sur la géométrie des solitons de Kähler-Ricci dans les variétés toriques et horosphériques." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS084/document.
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Cette thèse traite des solitons de Kähler-Ricci qui sont des généralisations naturelles des métriques de Kähler-Einstein. Elle est divisée en deux parties. La première étudie la décomposition solitonique de l’espace des champs de vecteurs holomorphes dans le cas des variétés toriques. La seconde partie étudie de manière analytique les variétés horosphériques en redémontrant par la méthode de la continuité l’existence de solitons de Kähler-Ricci sur ces variétés et en calculant après la borne supérieure de RicciThis thesis deal with Kähler-Ricci solitons which are natural generalizations of Kähler-Einstein metrics. It is divided into two parts. The first one studies the solitonic decomposition of the space of holomorphic vector spaces in the case of toric manifold. The second one studies is an analytic way the existence of horospherical Kähler-Ricci solitons on those manifolds and then computes the greatest Ricci lower bound
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Ozgoren, Kivanc. "Godel'." Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/12606497/index.pdf.
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In this thesis, firstly the original Gödel'
s metric is examined in detail. Then a more general class of Gö
del-type metrics is introduced. It is shown that they are the solutions of Einstein field equations with a physically acceptable matter distribution provided that some conditions are satisfied. Lastly, some examples of the Gö
del-type metrics are given.
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Santos, Alex Sandro Lopes. "Problema de Yamabe modificado em variedades compactas de dimensão quatro e métricas críticas do funcional curvatura escalar." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/22885.
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SANTOS, A. S. L. Problema de Yamabe modificado em variedades compactas de dimensão quatro e métricas críticas do funcional curvatura escalar. 2017. 58 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-25T19:34:47Z No. of bitstreams: 1 2017_tese_aslsantos.pdf: 535461 bytes, checksum: 8c3ddbdd33d74c4eb7b265354b3bafb3 (MD5)
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In the fisrt part of this work we investigate the modified Yamabe problem on four-dimensional manifolds whose the modifiers invariants depending on the eigenvalues of the Weyl curvature tensor and they are described in terms of maximum and minimum of the biorthogonal (sectional) curvature. We provide some geometrical and topological properties on four-dimensional manifolds in terms of these invariants. In the second part we investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980’s that every CPE metric must be Einstein. We prove that such a conjecture is true under a second-order vanishing condition on the Weyl tensor.
Na primeira parte deste trabalho investigamos o problema de Yamabe modificado em variedades de dimensão quatro cujos invariantes modificadores dependem dos autovalores do tensor de Weyl e são descritos em termos do máximo e mínimo da curvatura biortogonal (seccional). Fornecemos algumas propriedades geométricas e topológicas para tais variedades em termos destes invariantes. Na segunda parte investigamos os pontos críticos do funcional curvatura escalar total restrito ao espaço de métricas com curvatura escalar constante e volume unitário, abreviadamente chamamos de métricas CPE. Conjecturou-se na década de 1980 que toda métrica CPE deve ser Einstein. Provamos que tal conjectura é verdadeira sob uma condição de nulidade sobre o divergente de segunda ordem do tensor de Weyl.
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Da, Silva Caroline Dos Santos. "Cosmic strings and scalar tensor gravity." Thesis, Durham University, 1999. http://etheses.dur.ac.uk/4577/.
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This thesis is concerned with the study of cosmic strings. We studied the values for the Higgs mass and string coupling for which the gravitational effect of an infinite cosmic string in the context of the Einstein theory is not only locally but also globally weak. We conclude this happens for strings formed at scales less or equal to the Planck one with Higgs mass being less or equal to the boson vectorial mass. Then we examined the metric of an isolated self-gravitating abelian-Higgs vortex in dilatonic gravity for arbitrary coupling of the vortex fields to the dilaton. We looked for solutions in both massless and massive dilaton gravity. We compared our results to existing metrics for strings in Einstein and .Jordan-Brans-Dicke theories. We explored the generalisation of Bogomolnyi arguments for our vortices and commented on the effects on test particles. We then included the presence of an axion field and examined the metric of an isolated self-gravitating axionic-dilatonic string. Finally we studied dilatonic strings through black hole solutions in string theory. We concluded that the horizon of non-extreme charged black holes supports the long-range fields of the Nielsen-Olesen string that can be considered as black hole hair and whose gravitational effect is in general the production of a conical deficit into the metric of the black hole background. We also concluded that the effect of the dilaton on the horizon of these black holes is to generate an additional charge.
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Montefiori, Samuele. "Onde gravitazionali - teoria e rivelazione." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10334/.
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Nel redarre la tesi si è perseguito l'intento di illustrare la teoria alla base delle onde gravitazionali e dei metodi che ne consentono la rivelazione. È bene tenere presente che con il seguente elaborato non si sta proponendo, in alcun modo, una lettura da sostituire ad un testo didattico. Pur tuttavia, si è cercato di presentare gli argomenti in maniera tale da emulare l'itinerario formativo di uno studente che, per la prima volta, si approcci alle nozioni, non immediatamente intuitive, ivi descritte. Quindi, ogni capitolo è da interpretarsi come un passo verso la comprensione dei meccanismi fisici che regolano produzione, propagazione ed infine rivelazione delle perturbazioni di gravità.
Dopo una concisa introduzione, il primo capitolo si apre con il proposito di riepilogare i concetti basilari di geometria differenziale e relatività generale, gli stessi che hanno portato Einstein ad enunciare le famose equazioni di campo. Nel secondo si introduce, come ipotesi di lavoro standard, l'approssimazione di campo debole. Sotto questa condizione al contorno, per mezzo delle trasformazioni dello sfondo di Lorentz e di gauge, si manipolano le equazioni di Einstein, ottenendo la legge di gravitazione universale newtoniana.
Il terzo capitolo sfrutta le analogie tra equazioni di campo elettromagnetiche ed einsteiniane, mostrando con quanta naturalezza sia possibile dedurre l'esistenza delle onde gravitazionali. Successivamente ad averne elencato le proprietà, si affronta il problema della loro propagazione e generazione, rimanendo sempre in condizioni di linearizzazione. È poi la volta del quarto ed ultimo capitolo. Qui si avvia una dissertazione sui processi che acconsentono alla misurazione delle ampiezze delle radiazioni di gravità, esibendo le idee chiave che hanno condotto alla costruzione di interferometri all'avanguardia come LIGO. Il testo termina con uno sguardo alle recenti scoperte e alle aspettative future.
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Sebastianutti, Marco. "Geodesic motion and Raychaudhuri equations." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/18755/.
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The work presented in this thesis is devoted to the study of geodesic motion in the context of General Relativity. The motion of a single test particle is governed by the geodesic equations of the given space-time, nevertheless one can be interested in the collective behavior of a family (congruence) of test particles, whose dynamics is controlled by the Raychaudhuri equations. In this thesis, both the aspects have been considered, with great interest in the latter issue. Geometric quantities appear in these evolution equations, therefore, it goes without saying that the features of a given space-time must necessarily arise. In this way, through the study of these quantities, one is able to analyze the given space-time. In the first part of this dissertation, we study the relation between geodesic motion and gravity. In fact, the geodesic equations are a useful tool for detecting a gravitational field. While, in the second part, after the derivation of Raychaudhuri equations, we focus on their applications to cosmology. Using these equations, as we mentioned above, one can show how geometric quantities linked to the given space-time, like expansion, shear and twist parameters govern the focusing or de-focusing of geodesic congruences. Physical requirements on matter stress-energy (i.e., positivity of energy density in any frame of reference), lead to the various energy conditions, which must hold, at least in a classical context. Therefore, under these suitable conditions, the focusing of a geodesics "bundle", in the FLRW metric, bring us to the idea of an initial (big bang) singularity in the model of a homogeneous isotropic universe. The geodesic focusing theorem derived from both, the Raychaudhuri equations and the energy conditions acts as an important tool in understanding the Hawking-Penrose singularity theorems.
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RAFFERO, ALBERTO. "Non-integrable special geometric structures in dimensions six and seven." Doctoral thesis, 2016. http://hdl.handle.net/2318/1557510.
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Six-dimensional manifolds admitting an SU(3)-structure and seven-dimensional manifolds endowed with a G2-structure are the main object of study in this thesis. In the six-dimensional case, we consider SU(3)-structures (ω,ψ) satisfying the condition dω = c ψ, c ∈ R − {0}, known in literature as coupled. They are half-flat and generalize the class of nearly Kähler SU(3)-structures. We study their properties in the general case and in relation with the rôle they play in supersymmetric string theory, the conditions under which the associated metric is Einstein, their behaviour with respect to the Hitchin flow equations and various classes of examples. In the seven-dimensional case, we focus on G2-structures defined by a stable 3-form φ which is locally conformal equivalent to a closed one. We study the restrictions arising when the underlying metric is Einstein, we use warped products and the mapping torus construction to provide noncompact and compact examples of 7-manifolds endowed with such a structure starting from 6-manifolds with a coupled SU(3)-structure and, finally, we prove a structure result for compact 7-manifolds. We conclude studying a generalization of the Hitchin flow equations and a geometric flow of spinors on 6-manifolds. The latter gives rise to a flow of SU(3)-structures.
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Robles, Colleen. "Einstein metrics of Randers type." Thesis, 2003. http://hdl.handle.net/2429/14764.
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This thesis presents a study of Einstein Randers metrics. Initially introduced
within the context of relativity, Randers metrics have a strong
presence in both the theory and applications of Finsler geometry. The starting
point is a new characterization of Einstein metrics of Randers type by
three conditions. The conditions form a coupled, highly non-linear (due to
the presence of a Riemannian Ricci tensor), second order system of partial
differential equations. The equations are polynomial in the unknowns; a
Riemannian metric ã and differential 1-form b.
Recently Z. Shen has generalized Zermelo's problem of navigation on the
plane to arbitrary Riemannian manifolds. (The goal is to identify the paths
of shortest time on a Riemannian manifold (M, ă) under the influence of an
external force W = Wi∂xi.) In this context, Randers metrics may be viewed
as solutions to Zermelo's problem. The navigation structure yields the main
result of the thesis, a succinct geometric description of Einstein metrics
of Randers type. Explicitly, the Randers metric arising as the solution to
Zermelo's problem on (ă, W) is Einstein if and only if the Riemannian metric
ă is Einstein itself, and W is an infinitesimal homothety of ă.
The navigation description quickly yields a Schur lemma for the Ricci
curvature of Randers metrics. It is a testament to the navigation description
that this result, the first Schur lemma for Ricci curvature in (non-
Riemannian) Finsler geometry, is obtained with relative ease. An extension
of Matsumoto's Identity for Randers metrics of constant flag curvature to
the Einstein setting then follows.
Having established these general results, I then explore three scenarios:
Einstein metrics on surfaces of revolution, constant flag curvature metrics,
and Einstein metrics on closed manifolds. The thesis closes with a collection
of open questions.
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Sun, Jian. "Kähler-Einstein metrics and Sobolev inequality /." 2000. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:9965165.
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Pešta, Milan. "Prostoročasy prstencových zdrojů." Master's thesis, 2019. http://www.nusl.cz/ntk/nusl-393751.
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Marginally outer-trapped surfaces (MOTSs) are found for a family of space-like hypersurfaces described by the Brill-Lindquist initial data. These hypersurfaces contain a singular ring characterized by its radius, mass and charge. Due to the ring character of the singularity, these surfaces are natural candidates for MOTSs with toroidal topology. By adjusting and employing the numerical method of geodesics, we indeed localize MOTSs of both spherical and toroidal topology, and compare the results with those obtained previously by Jaramillo & Lousto.
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"Metricas de Einstein em variedades bandeira." Tese, Biblioteca Digital da Unicamp, 2005. http://libdigi.unicamp.br/document/?code=vtls000366713.
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Χρυσικός, Ιωάννης. "Ομογενείς μετρικές Einstein σε γενικευμένες πολλαπλότητες σημαιών." Thesis, 2011. http://nemertes.lis.upatras.gr/jspui/handle/10889/4418.
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Μια πολλαπλότητα Riemann (M, g) ονομάζεται Einstein αν έχει σταθερή καμπυλότητα Ricci.
Είναι γνωστό ότι αν (M=G/K, g) είναι μια συμπαγής ομογενής πολλαπλότητα Riemann,
τότε οι G-αναλλοίωτες μετρικές Einstein μοναδιαίου όγκου,
είναι τα κρίσιμα σημεία του συναρτησοειδούς ολικής βαθμωτής καμπυλότητας
περιορισμένο στο χώρο των G-αναλλοίωτων μετρικών με όγκο 1.
Για μια G-αναλλοίωτη μετρική Riemann η εξίσωση Einstein
ανάγεται σε ένα σύστημα αλγεβρικών εξισώσεων.
Οι θετικές πραγματικές λύσεις του συστήματος αυτού είναι
ακριβώς οι G-αναλλοίωτες μετρικές Einstein που δέχεται η
πολλαπλότητα Μ.
Μια σημαντική οικογένεια συμπαγών ομογενών χώρων αποτελείται
από τις γενικευμένες πολλαπλότητες σημαιών. Κάθε τέτοιος χώρος
είναι μια τροχιά της συζυγούς αναπαράστασης μιας συμπαγούς, συνεκτικής,
ημι-απλής ομάδας Lie G. Πρόκειται για ομογενείς πολλαπλότητες της
μορφής G/C(S), όπου C(S) είναι ο κεντροποιητής ενός δακτυλίου S στην G.
Κάθε τέτοιος χώρος δέχεται ένα πεπερασμένο πλήθος από
G-αναλλοίωτες μετρικές Kahler-EInstein.
Στην παρούσα διατριβή ταξινομούμε όλες τις πολλαπλότητες σημαιών
G/K που αντιστοιχούν σε μια απλή ομάδα Lie G,
των οποίων η ισοτροπική αναπαράσταση διασπάται σε 2 ή 4
μη αναγώγιμους και μη ισοδύναμους Ad(K)-αναλλοίωτους προσθετέους.
Για κάθε τέτοιο χώρο λύνουμε την αναλλοίωτη εξίσωση Εinstein,
και παρουσιάζουμε την αναλυτική μορφή νέων G-αναλλοίωτων μετρικών
Einstein. Στις περισσότερες περιπτώσεις παρουσιάζουμε την πλήρη ταξινόμηση των αναλλοίωτων μετρικών Einstein. Επίσης εξετάζουμε το ισομετρικό πρόβλημα.
Για την κατασκευή της εξίσωσης Einstein σε κάποιες
πολλαπλότητες σημαιών με 4 ισοτροπικούς προσθετέους
χρησιμοποιούμε την νηματοποίηση συστροφής που δέχεται
κάθε πολλαπλότητα σημαιών επί ενός ισοτροπικά
μη αναγώγιμου συμμετρικού χώρου συμπαγούς τύπου.
Αυτή η μέθοδος είναι καινούργια και μπορεί να εφαρμοστεί και σε άλλες πολλαπλότητες σημαιών.A Riemannian manifold (M, g) is called Einstein, if it has constant Ricci curvature. It is well known that if (M=G/K, g) is a compact homogeneous Riemannian manifold, then the G-invariant \tl{Einstein} metrics of unit volume, are the critical points of the scalar curvature function restricted to the space of all G-invariant metrics with volume 1. For a G-invariant Riemannian metric the Einstein equation reduces to a system of algebraic equations. The positive real solutions of this system are the $G$-invariant Einstein metrics on M. An important family of compact homogeneous spaces consists of the generalized flag manifolds. These are adjoint orbits of a compact semisimple Lie group. Flag manifolds of a compact connected semisimple Lie group exhaust all compact and simply connected homogeneous Kahler manifolds and are of the form G/C(S), where C(S) is the centralizer (in G) of a torus S in G. Such homogeneous spaces admit a finite number of G-invariant complex structures, and for any such complex structure there is a unique compatible G-invariant Kahler-Einstein metric. In this thesis we classify all flag manifolds M=G/K of a compact simple Lie group G, whose isotropy representation decomposes into 2 or 4, isotropy summands. For these spaces we solve the (homogeneous) Einstein equation, and we obtain the explicit form of new G-invariant Einstein metrics. For most cases we give the classification of homogeneous Einstein metrics. We also examine the isometric problem. For the construction of the Einstein equation on certain flag manifolds with four isotropy summands, we apply for first time the twistor fibration of a flag manifold over an isotropy irreducible symmetric space of compact type. This method is new and it can be used also for other flag manifolds. For flag manifolds with two isotropy summands, we use the restricted Hessian and we characterize the new Einstein metrics as local minimum points of the scalar curvature function restricted to the space of G-invariant Riemannian metrics of volume 1. We mention that the classification of flag manifolds with two isotropy summands gives us new examples of homogeneous spaces, for which the motion of a charged particle under the electromagnetic field, and the geodesics curves, are completely determined.
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Stemmler, Matthias [Verfasser]. "Stability and Hermitian-Einstein metrics for vector bundles on framed manifolds / vorgelegt von Matthias Stemmler." 2009. http://d-nb.info/1003965474/34.
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Σταθά, Μαρίνα. "Μελέτη γεωμετρίας σφαιρών και πολλαπλοτήτων Stiefel." Thesis, 2013. http://hdl.handle.net/10889/7985.
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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.
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Mujtaba, Abid Hasan. "Homogeneous Einstein Metrics on SU(n) Manifolds, Hoop Conjecture for Black Rings, and Ergoregions in Magnetised Black Hole Spacetimes." Thesis, 2013. http://hdl.handle.net/1969.1/149262.
Full textAbstract:
This Dissertation covers three aspects of General Relativity: inequivalent Einstein metrics on Lie Group Manifolds, proving the Hoop Conjecture for Black Rings, and investigating ergoregions in magnetised black hole spacetimes. A number of analytical and numerical techniques are employed to that end.
It is known that every compact simple Lie Group admits a bi-invariant homogeneous Einstein metric. We use two ansatze to probe the existence of additional inequivalent Einstein metrics on the Lie Group SU (n). We provide an explicit construction of 2k + 1 and 2k inequivalent Einstein metrics on SU (2k) and SU (2k + 1) respectively.
We prove the Hoop Conjecture for neutral and charged, singly and doubly rotating black rings. This allows one to determine whether a rotating mass distribution has an event horizon, that it is in fact a black ring.
We investigate ergoregions in magnetised black hole spacetimes. We show that, in general, rotating charged black holes (Kerr-Newman) immersed in an external magnetic field have ergoregions that extend to infinity near the central axis unless we restrict the charge to q = amB and keep B below a maximal value. Additionally, we show that as B is increased from zero the ergoregion adjacent to the event horizon shrinks, vanishing altogether at a critical value, before reappearing and growing until it is no longer bounded as B becomes greater than the maximal value.
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Kubeka, Amos Soweto. "Linear perturbations of a Schwarzschild black hole." Thesis, 2009. http://hdl.handle.net/10500/1667.
Full textAbstract:
We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric
space-times (originally calculated by Chandrasekhar) and we find some discrepancies
both in the linear and non-linear terms. However, these discrepancies do not affect
the results concerning linear perturbations of a Schwarzschild black hole. Secondly,
we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use
the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the
relation between these equations because they describe the same linear perturbations
of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular
momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum
equations obtained from the Bondi-Sachs metric and the Zerilli equation, because
they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by
means of a coordinate transformation, we extend Chandrasekhar's results on linear
perturbations of a Schwarzschild black hole to the Bondi-Sachs framework.Mathematical Sciences
M. Sc. (Applied Mathematics)