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1
Vilatte, Matthieu. "Adventures in (thermal) Wonderland". Electronic Thesis or Diss., Institut polytechnique de Paris, 2024. https://theses.hal.science/tel-04791687.
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Le travail que nous présentons dans cette thèse est structuré autour de la notion de théorie des champs et de géométrie, qui sont appliquées à la gravité et la thermalisation.En gravité, notre travail donne un éclairage nouveau sur la structure asymptotique du champ gravitationnel dans le contexte des espace-temps asymptotiquement plats, ceci en utilisant l'information codée sur leur bord conforme. Ce dernier est une hypersurface de genre lumière sur laquelle émerge la physique carrollienne au lieu de la physique relativiste. Une structure carrollienne sur une variété est constituée une métrique dégénérée et un champ de vecteurs couvrant le noyau de cette dernière. Ce vecteur sélectionne une direction particulière qui peut être le point de départ de la description des structures carrolliennes dans un cadre séparé. Nous développons d'abord la géométrie carrollienne, y compris une étude complète des connexions et isométries (conformes). Des actions effectives peuvent vivre sur un arrière-plan carrollien. Les moments canoniques conjugués à la géométrie ou à la connexion peuvent être définis, et la variation de l'action donnera leurs équations de conservation, à partir desquelles les charges isométriques peuvent être bâties.La physique carrollienne émerge également lorsque la vitesse de la lumière tend vers zéro. Cette limite donne généralement plus de descendants carrolliens que ce qui est attendu après une analyse intrinsèque, comme le montrent les exemples explicites des fluides carrolliens, des champs scalaires carrolliens (pour lesquels deux actions, électrique et magnétique, apparaissent dans la limite) et du tenseur de Cotton carrollien. La richesse de la limite est due à sa possibilité de décrire plus de degrés de liberté, ce qui s'avère être un outil fondamental dans l'étude de la relation entre les espace-temps asymptotiquement anti de Sitter et plats.Les espace-temps asymptotiquement plats peuvent être écrits comme une expansion infinie dans une jauge covariante par rapport à leur bord nul. Cette légère extension de la jauge de Newman-Unti est également valable dans AdS, ce qui permet de prendre la limite plate dans le bulk, équivalente à la limite carrollienne sur le bord. Nous démontrons que l'espace des solutions infini des espace-temps Ricci-plat provient en fait du développement en série de Laurent du tenseur énergie-impulsion d'AdS. Ces répliques obéissent à chaque ordre une dynamique carrollienne (lois de flux). Dans le cadre des espaces algébriquement spéciaux de Petrov (pour lesquels le développement infinie se resomme), nous utilisons les lois de flux carrolliennes ainsi que la conservation des tenseurs énergie-impulsion et de Cotton pour construire, du point de vue du bord, deux tours duales de charges du bulk. Parmi elles, nous retrouvons l'expansion mutipolaire de la masse et du moment angulaire pour la famille Kerr-Taub-NUT. La jauge covariante est également le cadre approprié pour dévoiler l'action des symétries cachées de la gravité sur le bord nul. Dans ce travail, nous étudions le cas de la symétrie SL(2,R) d'Ehlers.Du côté de la théorie thermique des champs, nous travaillons sur l'ensemble minimal de données nécessaires pour les décrire à température finie. Alors qu'à température infinie toutes les valeurs moyennes des opérateurs primaires s’annulent, leurs valeurs non nulle dans le cas thermique constituent les données supplémentaires qu'il faut calculer pour caractériser la théorie. Les simulations numériques, la dualité avec un trou noir dans AdS ou une analyse spectrale sont généralement les méthodes employées pour trouver la valeur de ces coefficients. Notre travail propose une nouvelle approche à ce problème en montrant, à partir de deux oscillateurs harmoniques couplés, que ces coefficients sont en fait liés à des graphes conformes de théories de type fishnet. A partir de cette observation, nous avons établi une correspondance entre les fonctions de partition thermique et ces graphesThe work we present in this thesis is structured around the concepts of field theories and geometry, which are applied to gravity and thermalisation.On the gravity side, our work aims at shedding new light on the asymptotic structure of the gravitational field in the context of asymptotically flat spacetimes, using information encoded on the conformal boundary. The latter is a null hypersurface on which Carrollian physics instead of relativistic physics is at work. A Carroll structure on a manifold is a degenerate metric and a vector field spanning the kernel of the latter. This vector selects a particular direction which can be the starting point for describing Carroll structures in a split frame. We first elaborate on the geometry one can construct on such a manifold in this frame, including a comprehensive study of connections and (conformal isometries). Effective actions can be defined on a Carrollian background. Canonical momenta conjugate to the geometry or the connection are introduced, and the variation of the action shall give their conservation equations, upon which isometric charges can be reached.Carrollian physics is also known to emerge as the vanishing speed of light of relativistic physics. This limit usually exhibits more Carrollian descendants than what might be expected from a naive intrinsic analysis, as shown in the explicit examples of Carrollian fluids, Carrollian scalar fields (for which two actions, electric and magnetic arise in the limit) and the Carrollian Chern-Simons action. The richness of the limiting procedure is due to this versatility in describing a palette of degrees of freedom. This turns out to be an awesome tool in studying the relationship between asymptotically anti de Sitter (AdS) and flat spacetimes.Metrics on asymptotically flat spacetimes can be expressed as an infinite expansion in a gauge, covariant with respect to their null boundaries. This slight extension of the Newman-Unti gauge is shown to be valid also in AdS, which allows to take the flat limit in the bulk i.e. the Carrollian limit on the boundary, while preserving this covariance feature. We demonstrate that the infinite solution space of Ricci-flat spacetimes actually arises from the Laurent expansion of the AdS boundary energy-momentum tensor. These replicas obey at each order Carrollian dynamics (flux/balance laws). Focusing our attention to Petrov algebraically special spacetimes (for which the infinite expansion resums), we use the Carrollian flux/balance laws together with the conservation of the energy-momentum and Cotton tensors to build two dual towers of bulk charges from a purely boundary perspective. Among them we recover the mass and angular momentum mutipolar moments for the Kerr-Taub-NUT family. The covariant gauge is also the appropriate framework to unveil the action of hidden symmetries of gravity on the null boundary. In this thesis we study exhaustively the case of Ehlers' $SL(2,mathbb{R})$ symmetry.On the side of thermal field theory we see that while at infinite temperature a CFT is described by its spectrum and the OPE coefficients, additional data is needed in the thermal case. These are the average values of primary operators, completely determined up to a constant coefficient. Numerical simulations, duality with black-hole states in AdS or spectral analyses are the methods usually employed to uncover the latter. Our work features a new breadth. Starting from two coupled harmonic oscillators, we show that they are related to conformal ladder graphs of fishnet theories. This observation is the first step for setting a new correspondence between thermal partition functions and graphs
2
Iakovidis, Nikolaos. "Geometry of Toric Manifolds". Thesis, Uppsala universitet, Teoretisk fysik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-277709.
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This project is an overview of Hamiltonian geometry on Kahler manifolds and of Kähler reduction. In the first section we define complex manifolds, give their basic properties and build some structures on them. We are mainly interested in Kähler manifolds which are a subset of symplectic manifolds. In the second section we discuss group actions on manifolds. We are only concerned with Hamiltonian actions for which we can compute their moment maps. From these we prove how to construct new manifolds using a process called Kähler reduction. Finally we define toric manifolds and review Delzant polytopes and their corresponding manifolds. In the third section we present detailed examples in order to clarify all previously given definitions.
3
Buttler, Michael. "The geometry of CR manifolds". Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.312247.
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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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Clancy, Robert. "Spin(7)-manifolds and calibrated geometry". Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:c37748b3-674a-4d95-8abf-7499474abce3.
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In this thesis we study Spin(7)-manifolds, that is Riemannian 8-manifolds with torsion-free Spin(7)-structures, and Cayley submanifolds of such manifolds. We use a construction of compact Spin(7)-manifolds from Calabi–Yau 4-orbifolds with antiholomorphic involutions, due to Joyce, to find new examples of compact Spin(7)-manifolds. We search the class of well-formed quasismooth hypersurfaces in weighted projective spaces for suitable Calabi–Yau 4-orbifolds. We consider antiholomorphic involutions induced by the restriction of an involution of the ambient weighted projective space and we classify anti-holomorphic involutions of weighted projective spaces. We consider the moduli problem for Cayley submanifolds of Spin(7)-manifolds and show that there is a fine moduli space of unobstructed Cayley submanifolds. This result improves on the work of McLean in that we consider the global issues of how to patch together the local result of McLean. We also use the work of Kriegl and Michor on ‘convenient manifolds’ to show that this moduli space carries a universal family of Cayley submanifolds. Using the analysis necessary for the study of the moduli problem of Cayleys we find examples of compact Cayley submanifolds in any compact Spin(7)-manifold arising, using Joyce’s construction, from a suitable Calabi–Yau 4-orbifold with antiholomorphic involution. For the analysis to work, we need to show that a given Cayley submanifold is unobstructed. To show that particular examples of Cayley submanifolds are unobstructed, we relate the obstructions of complex surfaces in Calabi–Yau 4-folds as complex submanifolds to the obstructions as Cayley submanifolds.
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Pena, Moises. "Geodesics on Generalized Plane Wave Manifolds". CSUSB ScholarWorks, 2019. https://scholarworks.lib.csusb.edu/etd/866.
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A manifold is a Hausdorff topological space that is locally Euclidean. We will define the difference between a Riemannian manifold and a pseudo-Riemannian manifold. We will explore how geodesics behave on pseudo-Riemannian manifolds and what it means for manifolds to be geodesically complete. The Hopf-Rinow theorem states that,“Riemannian manifolds are geodesically complete if and only if it is complete as a metric space,” [Lee97] however, in pseudo-Riemannian geometry, there is no analogous theorem since in general a pseudo-Riemannian metric does not induce a metric space structure on the manifold. Our main focus will be on a family of manifolds referred to as a generalized plane wave manifolds. We will prove that all generalized plane wave manifolds are geodesically complete.
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Tievsky, Aaron M. "Analogues of Kähler geometry on Sasakian manifolds". Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/45349.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.Includes bibliographical references (p. 53-54).
A Sasakian manifold S is equipped with a unit-length, Killing vector field ( which generates a one-dimensional foliation with a transverse Kihler structure. A differential form a on S is called basic with respect to the foliation if it satisfies [iota][epsilon][alpha] = [iota][epsilon]d[alpha] = 0. If a compact Sasakian manifold S is regular, i.e. a circle bundle over a compact Kähler manifold, the results of Hodge theory in the Kahler case apply to basic forms on S. Even in the absence of a Kähler base, there is a basic version of Hodge theory due to El Kacimi-Alaoui. These results are useful in trying to imitate Kähler geometry on Sasakian manifolds; however, they have limitations. In the first part of this thesis, we will develop a "transverse Hodge theory" on a broader class of forms on S. When we restrict to basic forms, this will give us a simpler proof of some of El Kacimi-Alaoui's results, including the basic dd̄-lemma. In the second part, we will apply the basic dd̄-lemma and some results from our transverse Hodge theory to conclude (in the manner of Deligne, Griffiths, and Morgan) that the real homotopy type of a compact Sasakian manifold is a formal consequence of its basic cohomology ring and basic Kähler class.
by Aaron Michael Tievsky.
Ph.D.
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Kotschick, Dieter. "On the geometry of certain 4 - manifolds". Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236179.
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Viaggi, Gabriele [Verfasser]. "Geometry of random 3-manifolds / Gabriele Viaggi". Bonn : Universitäts- und Landesbibliothek Bonn, 2020. http://d-nb.info/1208764896/34.
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Baier, P. D. "Special Lagrangian geometry". Thesis, University of Oxford, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.365884.
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Chu, Casey. "The Geometry of Data: Distance on Data Manifolds". Scholarship @ Claremont, 2016. https://scholarship.claremont.edu/hmc_theses/74.
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The increasing importance of data in the modern world has created a need for new mathematical techniques to analyze this data. We explore and develop the use of geometry—specifically differential geometry—as a means for such analysis, in two parts. First, we provide a general framework to discover patterns contained in time series data using a geometric framework of assigning distance, clustering, and then forecasting. Second, we attempt to define a Riemannian metric on the space containing the data in order to introduce a notion of distance intrinsic to the data, providing a novel way to probe the data for insight.
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GATTI, ALICE. "Special almost-Kähler geometry of some homogeneous manifolds". Doctoral thesis, Università degli studi di Pavia, 2019. http://hdl.handle.net/11571/1292132.
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In this thesis we study metrics with special curvature properties on some homogeneous
almost-Kähler manifolds. More precisely, given a symplectic manifold (M, ω) equipped with
a compatible almost-complex structure J, we consider the homogeneous equation ρ = λω,
where ρ is the Chern-Ricci form of J, that we call speciality condition. In particular, we
focus on two classes of symplectic manifolds: symplectic T 2 -bundles over T 2 and adjoint
orbits of semisimple Lie groups.
Symplectic T 2 -bundles over T 2 are distributed in five classes. We prove that the ones
belonging to four of these classes admit a special (Chern-Ricci flat) locally homogeneous
compatible almost-complex structure, while the ones in the remaining class do not admit
Chern-Ricci flat locally homogeneous compatible almost-complex structures. It is an open
problem whether they admit non-locally homogeneous special compatible almost-complex
structures.
Adjoint orbits of semisimple Lie groups turn out to be naturally almost-Kähler mani-
folds endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined
almost-complex structure. We give explicit formulae for the Chern-Ricci form, the Hermi-
tian scalar curvature and the Nijenhuis tensor in terms of root data and we discuss the
speciality condition, which may be translated in terms of root data as well. Moreover, we
examine when compact quotients of these orbits are Kähler manifolds.
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Dunn, Corey. "Curvature homogeneous pseudo-Riemannian manifolds /". view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1188874491&sid=3&Fmt=2&clientId=11238&RQT=309&VName=PQD.
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Thesis (Ph. D.)--University of Oregon, 2006.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 146-147). Also available for download via the World Wide Web; free to University of Oregon users.
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Winslow, George H. "Classification of Compact 2-manifolds". VCU Scholars Compass, 2016. http://scholarscompass.vcu.edu/etd/4291.
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It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then observe that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we delve into algebraic topology to determine that the aforementioned surfaces are not homeomorphic to one another, and thus we can place each surface into exactly one of these equivalence classes.
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盧貴榮 e Kwai-wing Eric Lo. "Harmonic maps in Kähler geometry". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B31214368.
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Lo, Kwai-wing Eric. "Harmonic maps in Kähler geometry /". Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B18716684.
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Kemp, M. C. "Geometric Seifert 4-manifolds with aspherical bases". University of Sydney. Mathematics, 2005. http://hdl.handle.net/2123/702.
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Seifert fibred 3-manifolds were originally defined and classified by Seifert. Scott gives a survey of results connected with these classical Seifert spaces, in particular he shows they correspond to 3-manifolds having one of six of the eight 3-dimensional geometries (in the sense of Thurston). Essentially, a classical Seifert manifold is a S1-bundle over a 2-orbifold. More generally, a Seifert manifold is the total space of a bundle over a 2-orbifold with flat fibres. It is natural to ask if these generalised Seifert manifolds describe geometries of higher dimension. Ue has considered the geometries of orientable Seifert 4-manifolds (which have general fibre a torus). He proves that (with a finite number of exceptions orientable manifolds of eight of the 4-dimensional geometries are Seifert fibred. However, Seifert manifolds with a hyperbolic base are not necessarily geometric. In this paper, we seek to extend Ue's work to the non-orientable case. Firstly, we will show that Seifert spaces over an aspherical base are determined (up to fibre preserving homeomorphism) by their fundamental group sequence. Furthermore when the base is hyperbolic, a Seifert space is determined (up to fibre preserving homeomorphism) by its fundamental group. This generalises the work of Zieschang, who assumed the base has no reflector curves, the fibre was a torus and that a monodromy of a loop surrounding a cone point is trivial. Then we restrict to the 4 dimensional case and find necessary and sufficient conditions for Seifert 4 manifolds over hyperbolic or Euclidean orbifolds to be geometric in the sense of Thurston. Ue proved that orientable Seifert 4-manifolds with hyperbolic base are geometric if and only if the monodromies are periodic, and we will prove that we can drop the orientable condition. Ue also proved that orientable Seifert 4-manifolds with a Euclidean base are always geometric, and we will again show the orientable assumption is unnecessary.
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Swann, Andrew F. "Hyperkähler and quaternionic Kähler geometry". Thesis, University of Oxford, 1990. http://ora.ox.ac.uk/objects/uuid:bb301f35-25e0-445d-8045-65e402908b85.
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A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kähler. A similar result is proved for 8-manifolds. HyperKähler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kähler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kähler and hyperKähier quotient constructions and allows quaternionic Kähler geometry to be subsumed into the theory of hyperKähler manifolds. It is shown that the hyperKähler metrics that arise admit a certain type of SU(2)- action, possess functions which are Kähler potentials for each of the complex structures simultaneously and determine quaternionic Kähler structures via a variant of the moment map construction. Quaternionic Kähler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKähler metrics with circle actions on complex line bundles over Kähler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKähler metrics defined by Kronheimer, are shown to give rise to quaternionic Kähler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kähler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.
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Pennec, Xavier. "Statistical Computing on Manifolds for Computational Anatomy". Habilitation à diriger des recherches, Université de Nice Sophia-Antipolis, 2006. http://tel.archives-ouvertes.fr/tel-00633163.
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During the last decade, my main research topic was on medical image analysis, and more particularly on image registration. However, I was also following in background a more theoretical research track on statistical computing on manifolds. With the recent emergence of computational anatomy, this topic gained a lot of importance in the medical image analysis community. During the writing of this habilitation manuscript, I felt that it was time to present a more simple and uni ed view of how it works and why it can be important. This is why the usual short synthesis of the habilitation became a hundred pages long text where I tried to synthesizes the main notions of statistical computing on manifolds with application in registration and computational anatomy. Of course, this synthesis is centered on and illustrated by my personal research work.
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Calin, Ovidiu. "The missing direction and differential geometry on Heisenberg manifolds". Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ53779.pdf.
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Božin, Vladimir 1973. "Geometry of Ricci-flat Kähler manifolds and some counterexamples". Thesis, Massachusetts Institute of Technology, 2004. http://hdl.handle.net/1721.1/32243.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (leaves 61-64).
In this work, we study geometry of Ricci-flat Kähler manifolds, and also provide some counterexample constructions. We study asymptotic behavior of complete Ricci-flat metrics at infinity and consider a construction of approximate Ricci-flat metrics on quasiprojective manifolds with a divisor with normal crossings removed, by means of reducing torsion of a non-Kähler metric with the right volume form. Next, we study special Lagrangian fibrations using methods of geometric function theory. In particular, we generalize the method of extremal length and prove a generaliziation of the Teichmiiller theorem. We relate extremal problems to the existence of special Lagrangian fibrations in the large complex structure limit of Calabi-Yau manifolds. We proceed to some problems in the theory of minimal surfaces, disproving the Schoen-Yau conjecture and providing a first example of a proper harmonic map from the unit disk to a complex plane. In the end, we prove that the union closed set conjecture is equivalent to a strengthened version, giving a construction which might lead to a counterexample.
by Vladimir Božin.
Ph.D.
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Suleymanova, Asilya. "On the spectral geometry of manifolds with conic singularities". Doctoral thesis, Humboldt-Universität zu Berlin, 2017. http://dx.doi.org/10.18452/18420.
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Wir beginnen mit der Herleitung der asymptotischen Entwicklung der Spur des Wärmeleitungskernes, $\tr e^{-t\Delta}$, für $t\to0+$, wobei $\Delta$ der Laplace-Beltrami-Operator auf einer Mannigfaltigkeit mit Kegel-Singularitäten ist; dabei folgen wir der Arbeit von Brüning und Seeley. Dann untersuchen wir, wie die Koeffizienten der Entwicklung mit der Geometrie der Mannigfaltigkeit zusammenhängen, insbesondere fragen wir, ob die (mögliche) Singularität der Mannigfaltigkeit aus den Koeffizienten - und damit aus dem Spektrum des Laplace-Beltrami-Operators - abgelesen werden kann. In wurde gezeigt, dass im zweidimensionalen Fall ein logarithmischer Term und ein nicht lokaler Term im konstanten Glied genau dann verschwinden, wenn die Kegelbasis ein Kreis der Länge $2\pi$ ist, die Mannigfaltigkeit also geschlossen ist. Dann untersuchen wir wir höhere Dimensionen. Im vier-dimensionalen Fall zeigen wir, dass der logarithmische Term genau dann verschwindet, wenn die Kegelbasis eine
sphärische Raumform ist. Wir vermuten, dass das Verschwinden eines nicht lokalen Beitrags zum konstanten Term äquivalent ist dazu, dass die Kegelbasis die runde Sphäre ist; das kann aber bisher nur im zyklischen Fall gezeigt werden. Für geraddimensionale Mannigfaltigkeiten höherer Dimension und mit Kegelbasis von konstanter Krümmung zeigen wir weiter, dass der logarithmische Term ein Polynom in der Krümmung ist, das Wurzeln ungleich 1 haben kann, so dass erst das Verschwinden von mehreren Termen - die derzeit noch nicht explizit behandelt werden können - die Geschlossenheit der Mannigfaltigkeit zur Folge haben könnte.We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with one conic singularity, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley. Then we investigate how the terms in the expansion reflect the geometry of the manifold. Since the general expansion contains a logarithmic term, its vanishing is a necessary condition for smoothness of the manifold. It is shown in the paper by Bruening and Seeley that in the two-dimensional case this implies that the constant term of the expansion contains a non-local term that determines the length of the (circular) cross section and vanishes precisely if this length equals $2\pi$, that is, in the smooth case. We proceed to the study of higher dimensions. In the four-dimensional case, the logarithmic term in the expansion vanishes precisely when the cross section is a spherical space form, and we expect that the vanishing of a further singular term will imply again smoothness, but this is not yet clear beyond the case of cyclic space forms. In higher dimensions the situation is naturally more difficult. We illustrate this in the case of cross sections with constant curvature. Then the logarithmic term becomes a polynomial in the curvature with roots that are different from 1, which necessitates more vanishing of other terms, not isolated so far.
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Lord, Steven. "Riemannian non-commutative geometry /". Title page, abstract and table of contents only, 2002. http://web4.library.adelaide.edu.au/theses/09PH/09phl8661.pdf.
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Botros, Amir A. "GEODESICS IN LORENTZIAN MANIFOLDS". CSUSB ScholarWorks, 2016. https://scholarworks.lib.csusb.edu/etd/275.
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We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Manifolds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between ``events''. They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called ``geodesically complete'', or simply, ``complete''. It is easy to show that the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the notions of spacelike complete, timelike complete, and null or lightlike complete. We illustrate that some of the notions are inequivalent.
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Herrera, Rafael. "Topics in geometry and topology". Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389011.
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Shen, Hongrui. "Unitarily invariant geometry on Grassmann manifold /". View abstract or full-text, 2006. http://library.ust.hk/cgi/db/thesis.pl?ECED%202006%20SHEN.
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Onodera, Mitsuko. "Study of rigidity problems for C2[pi]-manifolds". Sendai : Tohoku Univ, 2006. http://www.gbv.de/dms/goettingen/52860726X.pdf.
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Elmas, Gokhan. "Open book decompositions in high dimensional contact manifolds". Thesis, Georgia Institute of Technology, 2016. http://hdl.handle.net/1853/54967.
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In this thesis, we study the open book decompositions in high dimensional contact manifolds. We focus on the results about open book decomposition of manifolds and their relationship with contact geometry.
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Sepe, Daniele. "Integral affine geometry of Lagrangian bundles". Thesis, University of Edinburgh, 2011. http://hdl.handle.net/1842/5279.
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In this thesis, a bundle F →(M,ω) → B is said to be Lagrangian if (M,ω) is a 2n- dimensional symplectic manifold and the fibres are compact and connected Lagrangian submanifolds of (M,ω), i.e. ω |F = 0 for all F. This condition implies that the fibres and the base space are n-dimensional. Such bundles arise naturally in the study of a special class of dynamical systems in Hamiltonian mechanics, namely those called completely integrable Hamiltonian systems. A celebrated theorem due to Liouville [39], Mineur [46] and Arnol`d [2] provides a semi-global (i.e. in the neighbourhood of a fibre) symplectic classification of Lagrangian bundles, given by the existence of local action-angle coordinates. A proof of this theorem, due to Markus and Meyer [41] and Duistermaat [20], shows that the fibres and base space of a Lagrangian bundle are naturally integral affine manifolds, i.e. they admit atlases whose changes of coordinates can be extended to affine transformations of Rn which preserve the standard cocompact lattice Zn Rn. This thesis studies the problem of constructing Lagrangian bundles from the point of view of affinely at geometry. The first step to study this question is to construct topological universal Lagrangian bundles using the affine structure on the fibres. These bundles classify Lagrangian bundles topologically in the sense that every such bundle arises as the pullback of one universal bundle. However, not all bundles which are isomorphic to the pullback of a topological universal Lagrangian bundle are Lagrangian, as there exist further smooth and symplectic invariants. Even for bundles which admit local action-angle coordinates (these are classified up to isomorphism by topological universal Lagrangian bundles), there is a cohomological obstruction to the existence of an appropriate symplectic form on the total space, which has been studied by Dazord and Delzant in [18]. Such bundles are called almost Lagrangian. The second half of this thesis constructs the obstruction of Dazord and Delzant using the spectral sequence of a topological universal Lagrangian bundle. Moreover, this obstruction is shown to be related to a cohomological invariant associated to the integral affine geometry of the base space, called the radiance obstruction. In particular, it is shown that the integral a ne geometry of the base space of an almost Lagrangian bundle determines whether the bundle is, in fact, Lagrangian. New examples of (almost) Lagrangian bundles are provided to illustrate the theory developed.
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Lee, Hwasung. "Strominger's system on non-Kähler hermitian manifolds". Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:d3956c4f-c262-4bbf-8451-8dac35f6abef.
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In this thesis, we investigate the Strominger system on non-Kähler manifolds. We will present a natural generalization of the Strominger system for non-Kähler hermitian manifolds M with c₁(M) = 0. These manifolds are more general than balanced hermitian manifolds with holomorphically trivial canonical bundles. We will then consider explicit examples when M can be realized as a principal torus fibration over a Kähler surface S. We will solve the Strominger system on such construction which also includes manifolds of topology (k−1)(S²×S⁴)#k(S³×S³). We will investigate the anomaly cancellation condition on the principal torus fibration M. The anomaly cancellation condition reduces to a complex Monge-Ampère-type PDE, and we will prove existence of solution following Yau’s proof of the Calabi-conjecture [Yau78], and Fu and Yau’s analysis [FY08]. Finally, we will discuss the physical aspects of our work. We will discuss the Strominger system using α'-expansion and present a solution up to (α')¹-order. In the α'-expansion approach on a principal torus fibration, we will show that solving the anomaly cancellation condition in topology is necessary and sufficient to solving it analytically. We will discuss the potential problems with α'-expansion approach and consider the full Strominger system with the Hull connection. We will show that the α'-expansion does not correctly capture the behaviour of the solution even up to (α')¹-order and should be used with caution.
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Kemp, M. C. "Geometric Seifert 4-manifolds with aspherical bases". Thesis, The University of Sydney, 2005. http://hdl.handle.net/2123/702.
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Seifert fibred 3-manifolds were originally defined and classified by Seifert. Scott gives a survey of results connected with these classical Seifert spaces, in particular he shows they correspond to 3-manifolds having one of six of the eight 3-dimensional geometries (in the sense of Thurston). Essentially, a classical Seifert manifold is a S1-bundle over a 2-orbifold. More generally, a Seifert manifold is the total space of a bundle over a 2-orbifold with flat fibres. It is natural to ask if these generalised Seifert manifolds describe geometries of higher dimension. Ue has considered the geometries of orientable Seifert 4-manifolds (which have general fibre a torus). He proves that (with a finite number of exceptions orientable manifolds of eight of the 4-dimensional geometries are Seifert fibred. However, Seifert manifolds with a hyperbolic base are not necessarily geometric. In this paper, we seek to extend Ue's work to the non-orientable case. Firstly, we will show that Seifert spaces over an aspherical base are determined (up to fibre preserving homeomorphism) by their fundamental group sequence. Furthermore when the base is hyperbolic, a Seifert space is determined (up to fibre preserving homeomorphism) by its fundamental group. This generalises the work of Zieschang, who assumed the base has no reflector curves, the fibre was a torus and that a monodromy of a loop surrounding a cone point is trivial. Then we restrict to the 4 dimensional case and find necessary and sufficient conditions for Seifert 4 manifolds over hyperbolic or Euclidean orbifolds to be geometric in the sense of Thurston. Ue proved that orientable Seifert 4-manifolds with hyperbolic base are geometric if and only if the monodromies are periodic, and we will prove that we can drop the orientable condition. Ue also proved that orientable Seifert 4-manifolds with a Euclidean base are always geometric, and we will again show the orientable assumption is unnecessary.
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Leijon, Rasmus. "On the geometry of calibrated manifolds : with applications to electrodynamics". Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-80675.
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In this master thesis we study calibrated geometries, a family of Riemannian or Hermitian manifolds with an associated differential form, φ. We show that it isuseful to introduce the concept of proper calibrated manifolds, which are in asense calibrated manifolds where the geometry is derived from the calibration. In particular, the φ-Grassmannian is considered in the case of proper calibratedmanifolds. The impact of proper calibrated manifolds as a model is studied, aswell as the usefulness of pluripotential theory as tools for the model. The specialLagrangian calibration is an example of an important calibration introduced byHarvey and Lawson, which leads to the definition of the special Lagrangian differentialequation. This partial differential equation can be formulated in threeand four dimensions as det(H(u)) = Δu, where H(u) is the Hessian matrix of some potential u. We prove the existence of solutions and some other propertiesof this nonlinear differential equation and present the resulting 6- and 8-dimensional manifolds defined by the graph {x + i
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Schliebner, Daniel. "Contributions to the geometry of Lorentzian manifolds with special holonomy". Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2015. http://dx.doi.org/10.18452/17185.
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In dieser Arbeit studieren wir Lorentz-Mannigfaltigkeiten mit spezieller Holonomie, d.h. ihre Holonomiedarstellung wirkt schwach-irreduzibel aber nicht irreduzibel. Aufgrund der schwachen Irreduzibilität lässt die Darstellung einen ausgearteten Unterraum invariant und damit also auch eine lichtartige Linie. Geometrisch hat dies zur Folge, dass wir zwei parallele Unterbündel (die Linie und ihr orthogonales Komplement) des Tangentialbündels erhalten. Diese Arbeit nutzt diese und weitere Objekte um zu beweisen, dass kompakte Lorentzmannigfaltigkeiten mit Abelscher Holonomie geodätisch vollständig sind. Zudem werden Lorentzmannigfaltigkeiten mit spezieller Holonomie und nicht-negativer Ricci-Krümung auf den Blättern der Blätterung, induziert durch das orthogonale Komplement der parellelen Linie, und maximaler erster Bettizahl untersucht. Schließlich werden vollständige Ricci-flache Lorentzmannigfaltigkeiten mit vorgegebener voller Holonomie konstruiert.In the present thesis we study dimensional Lorentzian manifolds with special holonomy, i.e. such that their holonomy representation acts indecomposably but non-irreducibly. Being indecomposable, their holonomy group leaves invariant a degenerate subspace and thus a light-like line. Geometrically, this means that, since being holonomy invariant, this line gives rise to parallel subbundles of the tangent bundle. The thesis uses these and other objects to prove that Lorentian manifolds with Abelian holonomy are geodesically complete. Moreover, we study Lorentzian manifolds with special holonomy and non-negative Ricci curvature on the leaves of the foliation induced by the orthogonal complement of the parallel light-like line whose first Betti number is maximal. Finally, we provide examples of geodesically complete and Ricci-flat Lorentzian manifolds with special holonomy and prescribed full holonomy group.
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DI, NEZZA ELEONORA. "Geometry of complex Monge-Ampère equations on compact Kähler manifolds". Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2014. http://hdl.handle.net/2108/202161.
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Zuddas, Daniele. "Branched coverings and 4-manifolds". Doctoral thesis, Scuola Normale Superiore, 2007. http://hdl.handle.net/11384/85677.
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Sijbrandij, Klass Rienk. "The Toda equations and congruence in flag manifolds". Thesis, Durham University, 2000. http://etheses.dur.ac.uk/4516/.
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This thesis is concerned with the 2-dimensional Toda equations and their geometric interpretation in form of r-adapted maps into flag manifolds, r-adapted maps are not only of interest due to their relation with the Toda equations, but also for their adaption to the m-synametric space structure of flag manifolds. This thesis studies the congruence question for r-adapted maps in flag manifolds. The main theorem of this thesis is a congruence theorem for г-holomorphic maps Ψ : S(^2) → G/T of constant curvature, where G can be any compact simple Lie group. It is supplemented by a congruence theorem for general r-holomorphic maps Ψ : S(^2) → G/T if G has rank 2, and a number of congruence theorems for isometric r-primitive Ψ : S(^2) → G/T of constant Kahler angle. The second group of congruence theorems is proved for the rank 2 case, as well as a selection of Lie groups with higher rank: SU(4),SU(5),F(_4),E(_6),E(_6),E(_8),Sp(n).
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Di, Natale Carmelo. "Grassmannians and period mappings in derived algebraic geometry". Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.709191.
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Ma, Yilin. "Nonlinear Calderón Problem on Stein Manifolds". Thesis, The University of Sydney, 2021. https://hdl.handle.net/2123/25757.
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This thesis is devoted to the study of inverse problems for semilinear elliptic equations on Stein manifolds with Kähler metric. After developing some preliminary techniques, we will show that the Dirichlet-Neumann maps for certain semilinear elliptic equations determine the nonlinearities. We will consider two inverse problems of this kind with distinct geometric conditions imposed. The first one is the inverse problem for nonlinear Schrödinger equations on Kähler manifolds having specific Stein-like properties. The second one is the inverse problem for nonlinear magnetic Schrödinger equations on Riemann surfaces with partial data boundary measurements. In both cases, the nonlinearities involved are assumed to have certain analytic representations and vanishing lower order terms. The key observation is that, by a suitable linearisation procedure, one could transform the nonlinear problems into series of linear problems which have close connections to the techniques we develop.
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Fauck, Alexander. "Rabinowitz-Floer homology on Brieskorn manifolds". Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17501.
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In dieser Dissertation werden Kontaktstrukturen auf beliebigen differenzierbaren Mannigfaltigkeiten ungerader Dimension untersucht. Dies geschiet vermöge der Rabinowitz-Floer-Homologie (RFH), welche 2009 von Cieliebak und Frauenfelder eingeführt wurde. Ein großer Teil der Arbeit widmet sich den technischen Problemen bei der Definition von RFH. Insbesondere wird die Transversalität für die benötigten Modulräume gezeigt. In einem weiteren Abschnitt wird bewiesen, dass RFH im wesentlichen invariant unter subkrittischer Henkelanklebung ist. Schließlich enthält die Arbeit die Berechnung von RFH für einige Brieskorn-Mannigfaltigkeiten. Die dabei gewonnenen Resultate werden dazu verwendet zu zeigen, dass es auf jeder Mannigfaltigkeit, welche füllbare Kontaktstukturen zulässt, entweder unendlich viele verschiedene füllbare Kontaktstrukturen gibt, oder eine Kontaktstruktur mit unendlich vielen verschiedenen Füllungen oder das für alle füllbaren Kontaktstrukturen die RFH von unendlicher Dimension ist für alle Grade.This thesis considers fillable contact structures on odd-dimensional manifolds. For that purpose, Rabinowitz-Floer homology (RFH) is used which was introduced by Cieliebak and Frauenfelder in 2009. A major part of the thesis is devoted to technical problems in the definition of RFH. In particular, it is shown that the moduli spaces involved are cut out transversally. Moreover, it is proved that RFH is essentially invariant under subcritical handle attachment. Finally, RFH is calculated for some Brieskorn manifolds. The obtained results are then used to show for every manifold, which supports fillable contact structures, that there exist either infinitely many different fillable contact structures, or one contact structure with infinitely many different fillings or for every fillable contact structure holds that RFH is infinite dimensional in every degree.
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Miscione, Steven. "Loop algebras and algebraic geometry". Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=116115.
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This thesis primarily discusses the results of two papers, [Hu] and [HaHu]. The first is an overview of algebraic-geometric techniques for integrable systems in which the AKS theorem is proven. Under certain conditions, this theorem asserts the commutatvity and (potential) non-triviality of the Hamiltonian flow of Ad*-invariant functions once they're restricted to subalgebras. This theorem is applied to the case of coadjoint orbits on loop algebras, identifying the flow with a spectral curve and a line bundle via the Lax equation. These results play an important role in the discussion of [HaHu], wherein we consider three levels of spaces, each possessing a linear family of Poisson spaces. It is shown that there exist Poisson mappings between these levels. We consider the two cases where the underlying Riemann surface is an elliptic curve, as well as its degeneration to a Riemann sphere with two points identified (the trigonometric case). Background in necessary areas is provided.
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Masters, Joseph David. "Lengths and homology of hyperbolic 3-manifolds /". Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.
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Taft, Jefferson. "Intrinsic Geometric Flows on Manifolds of Revolution". Diss., The University of Arizona, 2010. http://hdl.handle.net/10150/194925.
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An intrinsic geometric flow is an evolution of a Riemannian metric by a two-tensor. An extrinsic geometric flow is an evolution of an immersion of a manifold into Euclidean space. An extrinsic flow induces an evolution of a metric because any immersed manifold inherits a Riemannian metric from Euclidean space. In this paper we discuss the inverse problem of specifying an evolution of a metric and then seeking an extrinsic geometric flow which induces the given metric evolution. We limit our discussion to the case of manifolds that are rotationally symmetric and embeddable with codimension one. In this case, we reduce an intrinsic geometric flow to a plane curve evolution. In the specific cases we study, we are able to further simplify the evolution to an evolution of a function of one variable. We provide soliton equations and give proofs that some soliton metrics exist.
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Kuhlmann, Sally Malinda. "Geodesic knots in hyperbolic 3 manifolds". Connect to thesis, 2005. http://repository.unimelb.edu.au/10187/916.
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This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds.Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded.
The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially.
We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.
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Arsie, Alessandro. "On "special" embeddings in complex and projective algebraic geometry". Doctoral thesis, SISSA, 2001. http://hdl.handle.net/20.500.11767/4302.
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Dilts, James. "The Einstein Constraint Equations on Asymptotically Euclidean Manifolds". Thesis, University of Oregon, 2015. http://hdl.handle.net/1794/19237.
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In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constraint equations for near-constant-mean-curvature (near-CMC) data as well as for far-from-CMC data, a proof of the limit equation criterion in the near-CMC case, as well as a model problem on the relationship between the asymptotic constants of solutions and the ADM mass. We also prove a characterization of the Yamabe classes on asymptotically Euclidean manifolds and resolve the (conformally) prescribed scalar curvature problem on asymptotically Euclidean manifolds for the case of nonpositive scalar curvatures.
This dissertation includes previously published coauthored material.
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Kazaras, Demetre. "Gluing manifolds with boundary and bordisms of positive scalar curvature metrics". Thesis, University of Oregon, 2017. http://hdl.handle.net/1794/22698.
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This thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest.
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Nordström, Johannes. "Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy". Thesis, University of Cambridge, 2008. https://www.repository.cam.ac.uk/handle/1810/214782.
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In Berger's classification of Riemannian holonomy groups there are several infinite families and two exceptional cases: the groups Spin(7) and G_2. This thesis is mainly concerned with 7-dimensional manifolds with holonomy G_2. A metric with holonomy contained in G_2 can be defined in terms of a torsion-free G_2-structure, and a G_2-manifold is a 7-dimensional manifold equipped with such a structure. There are two known constructions of compact manifolds with holonomy exactly G_2. Joyce found examples by resolving singularities of quotients of flat tori. Later Kovalev found different examples by gluing pairs of exponentially asymptotically cylindrical (EAC) G_2-manifolds (not necessarily with holonomy exactly G_2) whose cylinders match. The result of this gluing construction can be regarded as a generalised connected sum of the EAC components, and has a long approximately cylindrical neck region. We consider the deformation theory of EAC G_2-manifolds and show, generalising from the compact case, that there is a smooth moduli space of torsion-free EACG_2-structures. As an application we study the deformations of the gluing construction for compact G_2-manifolds, and find that the glued torsion-free G_2-structures form an open subset of the moduli space on the compact connected sum. For a fixed pair of matching EAC G_2-manifolds the gluing construction provides a path of torsion-free G_2-structures on the connected sum with increasing neck length. Intuitively this defines a boundary point for the moduli space on the connected sum, representing a way to 'pull apart' the compact G_2-manifold into a pair of EAC components. We use the deformation theory to make this more precise. We then consider the problem whether compact G_2-manifolds constructed by Joyce's method can be deformed to the result of a gluing construction. By proving a result for resolving singularities of EAC G_2-manifolds we show that some of Joyce's examples can be pulled apart in the above sense. Some of the EAC G_2-manifolds that arise this way satisfy a necessary and sufficient topological condition for having holonomy exactly G_2. We prove also deformation results for EAC Spin(7)-manifolds, i.e. dimension 8 manifolds with holonomy contained in Spin(7). On such manifolds there is a smooth moduli space of torsion-free EAC Spin(7)-structures. Generalising a result of Wang for compact manifolds we show that for EAC G_2-manifolds and Spin(7)-manifolds the special holonomy metrics form an open subset of the set of Ricci-flat metrics.
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Hines, Clinton M. "Spin Cobordism and Quasitoric Manifolds". UKnowledge, 2014. http://uknowledge.uky.edu/math_etds/17.
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This dissertation demonstrates a procedure to view any quasitoric manifold as a “minimal” sub-manifold of an ambient quasitoric manifold of codimension two via the wedge construction applied to the quotient polytope. These we term wedge quasitoric manifolds. We prove existence utilizing a construction on the quotient polytope and characteristic matrix and demonstrate conditions allowing the base manifold to be viewed as dual to the first Chern class of the wedge manifold. Such dualization allows calculations of KO characteristic classes as in the work of Ochanine and Fast. We also examine the Todd genus as it relates to two types of wedge quasitoric manifolds. Background matter on polytopes and toric topology, as well as spin and complex cobordism are provided.
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Remsing, Claidiu Cristian. "Tangentially symplectic foliations". Thesis, Rhodes University, 1994. http://hdl.handle.net/10962/d1005233.
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This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.
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Casey, Meredith Perrie. "Branched covers of contact manifolds". Diss., Georgia Institute of Technology, 2013. http://hdl.handle.net/1853/50313.
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We will discuss what is known about the construction of contact structures via branched covers, emphasizing the search for universal transverse knots. Recall that a topological knot is called universal if all 3-manifold can be obtained as a cover of the 3-sphere branched over that knot. Analogously one can ask if there is a transverse knot in the standard contact structure on S³ from which all contact 3-manifold can be obtained as a branched cover over this transverse knot. It is not known if such a transverse knot exists.