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Dissertations / Theses on the topic 'Congruences (Geometry) Geometry, Differential'

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Published: 4 June 2021

Last updated: 19 February 2022

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1

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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2

Whiteway, L. "Topics in differential geometry." Thesis, University of Oxford, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.379896.

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3

Ward, Thomas. "K1-congruences between L-values of elliptic curves." Thesis, University of Nottingham, 2009. http://eprints.nottingham.ac.uk/10766/.

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We study the L-values of an elliptic curve twisted by an Artin representation. Specifically, we consider the case in which the representation factors through a false Tate curve extension of Q. First, we consider a semistable elliptic curve E; we construct an integral-valued p-adic measure which interpolates the values the L-values of an Artin twist of E, at a family of finite-order character twists. To do this, we exploit the fact that such an L-value may be written as the Rankin convolution of two Hilbert modular forms, when the representation factors through the false Tate curve extension. Recent developments in non-abelian Iwasawa theory predict certain strong congruences between these p-adic L-functions, and we shall establish weakened versions of these congruences. Next, we prove analogous results for an elliptic curve with complex multiplication; we do this using work of Hida and Tilouine on the p-adic interpolation of Hecke L-functions over a CM-field. We go on to investigate the ratio of the automorphic and motivic periods associated to E in this setting. We describe how the p-valuation of this ratio may be explicitly calculated, and use the computer package MAGMA to produce some numerical examples. We end by proving a formula for the growth of this quantity in terms of the Iwasawa invariants associated to the two-variable extension of the CM-field.

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4

Taylor, Thomas E. "Differential geometry of Minkowski spaces." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp04/mq24990.pdf.

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5

Lenssen, Mark. "A topic in differential geometry." Thesis, University of Oxford, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.314920.

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6

Guo, Guang-Yuan. "Differential geometry of holomorphic bundles." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239283.

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7

Hale, Mark. "Developments in noncommutative differential geometry." Thesis, Durham University, 2002. http://etheses.dur.ac.uk/3948/.

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One of the great outstanding problems of theoretical physics is the quantisation of gravity, and an associated description of quantum spacetime. It is often argued that, at short distances, the manifold structure of spacetime breaks down and is replaced by some sort of algebraic structure. Noncommutative geometry is a possible candidate for the mathematics of this structure. However, physical theories on noncommutative spaces are still essentially classical and need to be quantised. We present a path integral formalism for quantising gravity in the form of the spectral action. Our basic principle is to sum over all Dirac operators. The approach is demonstrated on two simple finite noncommutative geometries (the two-point space and the matrix geometry M(_2)(C)) and a circle. In each case, we start with the partition function and calculate the graviton propagator and Greens functions. The expectation values of distances are also evaluated. We find on the finite noncommutative geometries, distances shrink with increasing graviton excitations, while on a circle, they grow. A comparison is made with Rovelli's canonical quantisation approach, and with his idea of spectral path integrals. We also briefly discuss the quantisation of a general Riemannian manifold. Included, is a comprehensive overview of the homological aspects of noncommutative geometry. In particular, we cover the index pairing between K-theory and K-homology, KK-theory, cyclic homology/cohomology, the Chern character and the index theorem. We also review the various field theories on noncommutative geometries.

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8

Bartocci, C. "Foundations of graded differential geometry." Thesis, University of Warwick, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386972.

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9

Zharkov, Sergei. "Conic structures in differential geometry." Thesis, Connect to e-thesis, 2000. http://theses.gla.ac.uk/1005/.

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Thesis (Ph.D.) -- University of Glasgow, 2000.<br>Includes bibliographical references (p.86-88). Print version also available. Mode of access : World Wide Web. System requirements : Adobe Acrobat reader required to view PDF document.

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10

Mazenc, Edward A. "Multifield inflation and differential geometry." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/83809.

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Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2013.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 63-67).<br>Cosmic inflation posits that the universe underwent a period of exponential expansion, driven by one or several quantum fields, shortly after the Big Bang. Renormalization requires the fields be non-minimally coupled to gravity. We examine such multifield models and find a rich geometric structure. After a conformal transformation of spacetime, the target field-space acquires non-trivial curvature. We explore two main consequences. First, we construct a field-space covariant framework to study quantum perturbations, extending prior work beyond the slow-roll approximation by working on the full phase space of the theory. Secondly, we show that a wide class of inflationary models can be understood as a geodesic motion on a suitably related manifold. Our geometric approach provides great insight into the (classical) field dynamics, and we have used them to compute non-gaussianities in the cosmic microwave background radiation spectrum.<br>by Edward A. Mazenc.<br>S.B.

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11

Lu, Adonis. "Statistical Theory Through Differential Geometry." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/cmc_theses/2181.

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This thesis will take a look at the roots of modern-day information geometry and some applications into statistical modeling. In order to truly grasp this field, we will first provide a basic and relevant introduction to differential geometry. This includes the basic concepts of manifolds as well as key properties and theorems. We will then explore exponential families with applications of probability distributions. Finally, we select a few time series models and derive the underlying geometries of their manifolds.

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12

Nash, Oliver. "Differential geometry of monopole moduli spaces." Thesis, University of Oxford, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.437029.

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13

Marriott, Paul. "Applications of differential geometry to statistics." Thesis, University of Warwick, 1990. http://wrap.warwick.ac.uk/55719/.

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Chapters 1 and 2 are both surveys of the current work in applying geometry to statistics. Chapter 1 is a broad outline of all the work done so far, while Chapter 2 studies, in particular, the work of Amari and that of Lauritzen. In Chapters 3 and 4 we study some open problems which have been raised by Lauritzen's work. In particular we look in detail at some of the differential geometric theory behind Lauritzen's defmition of a Statistical manifold. The following chapters follow a different line of research. We look at a new non symmetric differential geometric structure which we call a preferred point manifold. We show how this structure encompasses the work of Amari and Lauritzen, and how it points the way to many generalizations of their results. In Chapter 5 we define this new structure, and compare it to the Statistical manifold theory. Chapter 6 develops some examples of the new geometry in a statistical context. Chapter 7 starts the development of the pure theory of these preferred point manifolds. In Chapter 8 we outline possible paths of research in which the new geometry may be applied to statistical theory. We include, in an appendix, a copy of a joint paper which looks at some direct applications of differential geometry to a statistical problem, in this case it is the problem of the behaviour of the Wald test with nonlinear restriction functions.

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14

West, Janet Mary. "The differential geometry of the crosscap." Thesis, University of Liverpool, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260330.

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15

Davis, Declan Denis Daniel. "Affine differential geometry and singularity theory." Thesis, University of Liverpool, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.479061.

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16

Flari, Magdalini K. "Triple vector bundles in differential geometry." Thesis, University of Sheffield, 2018. http://etheses.whiterose.ac.uk/21385/.

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The triple tangent bundle T3M of a manifold M is a prime example of a triple vector bundle. The definition of a general triple vector bundle is a cube of vector bundles that commute in the strict categorical sense. We investigate the intrinsic features of such cubical structures, introducing systematic notation, and further studying linear double sections; a generalization of sections of vector bundles. A set of three linear double sections on a triple vector bundle E yields a total of six different routes from the base manifold M of E to the total space E. The underlying commutativity of the vector bundle structures of E leads to the concepts of warp and ultrawarp, concepts that measure the noncommutativity of the six routes. The main theorem shows that despite this noncommutativity, there is a strong relation between the ultrawarps. The methods developed to prove the theorem rely heavily on the analysis of the core double vector bundles and of the ultracore vector bundle of E. This theorem provides a conceptual proof of the Jacobi identity, and a new interpretation of the curvature of a connection on a vector bundle A.

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17

Hadnot, Jason. "Differential geometry of Fermat quartic surface." Thesis, Boston University, 2013. https://hdl.handle.net/2144/12773.

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Thesis (Ph.D.)--Boston University PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you.<br>We examine the differential geometry of the Fermat Quartic surface X4/0+X4/3-X4/1-X4/2 = 0 in CP^3 with induced Fubini-Study metric. We show that the differential equations of geodesics, when restricted to the real Fermat quartic surface inside the full complex quartic, can be reduced to two non-linear differential equations with rational coefficients along especially chosen geodesics. This simplification opens up the possibility of parametrizing these geodesics in terms of genus three Abelian integrals and their inversions. Furthermore the identity component of the differential Galois group of normal variational equation, derived from the geodesic equation along one of these selected curves, is SL(2,C). By Morales-Ramis theory the Hamiltonian system defining the geodesic equations is not integrable in a neighborhood of this solution by meromorphic integrals.

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18

Clarke, Daniel. "Integrability in submanifold geometry." Thesis, University of Bath, 2012. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.558890.

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This thesis concerns the relationship of submanifold geometry, in both the smooth and discrete sense, to representation theory and the theory of integrable systems. We obtain Lie theoretic generalisations of the transformation theory of projectively and Lie applicable surfaces, and M�obius-flat submanifolds of the conformal n-sphere. In the former case, we propose a discretisation. We develop a projective approach to centro-ane hypersurfaces, analogous to the conformal approach to submanifolds in spaceforms. This yields a characterisation of centro-ane hypersurfaces amongst M�obius-flat projective hypersurfaces using polynomial conserved quantities. We also propose a discretisation of curved flats in symmetric spaces. After developing the transformation theory for this, we see how Darboux pairs of discrete isothermicnets arise as discrete curved flats in the symmetric space of opposite point pairs. We show how discrete curves in the 2-sphere fit into this framework.

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19

Duran, James Joseph. "Differential geometry of surfaces and minimal surfaces." CSUSB ScholarWorks, 1997. https://scholarworks.lib.csusb.edu/etd-project/1542.

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20

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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21

Liu, Yang, and 劉洋. "Optimization and differential geometry for geometric modeling." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40988077.

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22

Bousfield, R. A. "Applications of differential geometry to structural mechanics." Thesis, University of Hertfordshire, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372544.

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23

Liu, Yang. "Optimization and differential geometry for geometric modeling." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40988077.

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24

Holanda, Felipe D'Angelo. "Introduction to differential geometry of plane curves." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15052.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior<br>A intenÃÃo desse trabalho serÃ de abordar de forma bÃsica e introdutÃria o estudo da Geometria Diferencial, que por sua vez tem seus estudos iniciados com as Curvas Planas. SerÃ necessÃrio um conhecimento de CÃlculo Diferencial, Integral e Geometria AnalÃtica para melhor compreensÃo desse trabalho, pois como seu prÃprio nome nos transparece Geometria Diferencial vem de uma junÃÃo do estudo da Geometria envolvendo CÃlculo. Assim abordaremos subtemas como curvas suaves, vetor tangente, comprimento de arco passando por fÃrmulas de Frenet, curvas evolutas e involutas e finalizaremos com alguns teoremas importantes, como o teorema fundamental das curvas planas, teorema de Jordan e o teorema dos quatro vÃrtices. O que, basicamente representa, o capÃtulo 1, 4 e 6 do livro IntroduÃÃo Ãs Curvas Planas de HilÃrio Alencar e Walcy Santos.<br>The intention of this work is to address in basic form and introductory study of Differential Geometry, which in turn has started his studies with Planas curves. It will require a knowledge of Differential Calculus, Integral and Analytic Geometry for better understanding of this work, because as its name says in Differential Geometry comes from the joint study of geometry involving Calculation. So we discuss sub-themes as smooth curves, tangent vector, arc length through formulas of Frenet, evolutas curves and involute and conclude with some important theorems, as the fundamental theorem of plane curves, Jordan 's theorem and the theorem of four vertices. What basically is, Chapter 1, 4 and 6 of the book Introduction to Plane Curves HilÃrio Alencar and Walcy Santos.

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25

Dangskul, Supreedee. "Construction of Seifert surfaces by differential geometry." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/20382.

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A Seifert surface for a knot in ℝ³ is a compact orientable surface whose boundary is the knot. Seifert surfaces are not unique. In 1934 Herbert Seifert provided a construction of such a surface known as the Seifert Algorithm, using the combinatorics of a projection of the knot onto a plane. This thesis presents another construction of a Seifert surface, using differential geometry and a projection of the knot onto a sphere. Given a knot K : S¹⊂ R³, we construct canonical maps F : ΛdiffS² → ℝ=4πZ and G : ℝ³ - K(S¹) → ΛdiffS² where ΛdiffS² is the space of smooth loops in S². The composite FG : ℝ³ - K(S¹) → ℝ=4πZ is a smooth map defined for each u∈2 ℝ³ - K(S¹) by integration of a 2- form over an extension D² → S² of G(u) : S1 → S². The composite FG is a surjection which is a canonical representative of the generator 1∈H¹(ℝ³- K(S¹)) = Z. FG can be defined geometrically using the solid angle. Given u ∈ ℝ³ - K(S¹), choose a Seifert surface Σu for K with u ∉ Σu. It is shown that FG(u) is equal to the signed area of the shadow of Σu on the unit sphere centred at u. With this, FG(u) can be written as a line integral over the knot. By Sard's Theorem, FG has a regular value t ∈ ℝ=4πZ. The behaviour of FG near the knot is investigated in order to show that FG is a locally trivial fibration near the knot, using detailed differential analysis. Our main result is that (FG)-¹(t)⊂ ℝ³ can be closed to a Seifert surface by adding the knot.

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26

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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27

Liu, Dunxue Carleton University Dissertation Mathematics. "Dihedral polynomial congruences and binary quadratic forms: a class field theory approach." Ottawa, 1992.

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28

Pinsky, Nathan. "Mathematical Knowledge for Teaching and Visualizing Differential Geometry." Scholarship @ Claremont, 2013. http://scholarship.claremont.edu/hmc_theses/49.

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In recent decades, education researchers have recognized the need for teachers to have a nuanced content knowledge in addition to pedagogical knowledge, but very little research was conducted into what this knowledge would entail. Beginning in 2008, math education researchers began to develop a theoretical framework for the mathematical knowledge needed for teaching, but their work focused primarily on elementary schools. I will present an analysis of the mathematical knowledge needed for teaching about the regular curves and surfaces, two important concepts in differential geometry which generalize to the advanced notion of a manifold, both in a college classroom and in an on-line format. I will also comment on the philosophical and political questions that arise in this analysis.

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29

McCormick, Andrew Grady. "Discrete Differential Geometry and Physics of Elastic Curves." Thesis, Harvard University, 2013. http://dissertations.umi.com/gsas.harvard:11121.

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30

Bakker, Craig Kent Reddick. "A differential geometry framework for multidisciplinary design optimization." Thesis, University of Cambridge, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.708688.

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31

Akıncı, Figen Pashaev Oktay K. "Geometry of moving curves and soliton equations/." [s.l.]: [s.n.], 2004. http://library.iyte.edu.tr/tezler/master/matematik/T000454.pdf.

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32

Kirchhoff-Lukat, Charlotte Sophie. "Aspects of generalized geometry : branes with boundary, blow-ups, brackets and bundles." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/283007.

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This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.

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33

Strawn, Nathaniel Kirk. "Geometry and constructions of finite frames." [College Station, Tex. : Texas A&M University, 2007. http://hdl.handle.net/1969.1/ETD-TAMU-1335.

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34

Botnan, Magnus Bakke. "Three Approaches in Computational Geometry and Topology : Persistent Homology, Discrete Differential Geometry and Discrete Morse Theory." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-14201.

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We study persistent homology, methods in discrete differential geometry and discrete Morse theory. Persistent homology is applied to computational biology and range image analysis. Theory from differential geometry is used to define curvature estimates of triangulated hypersurfaces. In particular, a well-known method for triangulated surfacesis generalised to hypersurfaces of any dimension. The thesis concludesby discussing a discrete analogue of Morse theory.

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Pitucco, Anthony Peter. "Differential-geometric aspects of adapted contact structures." Diss., The University of Arizona, 1991. http://hdl.handle.net/10150/185532.

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Let M denote a 2n-dimensional globally defined orientable manifold from which is constructed the product space N = M x R. It is assumed that N is endowed with a set of 2n independent smooth 1-forms {π(h),πʰ:h = 1,..,n}. Certain conditions are imposed on {π(h),πʰ} which imply the existence of local coordinates {qʰ,p(h)} on M and a function H(qʰ,p(h),t) on N, where t is the single coordinate on R, such that dπ = π(h) ∧ πʰ, where π has the structure of a Cartan form on N. The assumption that the function h = p(h)∂H/∂p(h)-H is non-zero on a region D ⊂ N, implies that π has maximal class on D. This construction gives rise to a local adapted contact structure on N and a local symplectic structure on M. A vector field X on N is said to be a contact field if there exists a smooth function σ : N → R such that ₤ₓπ = σπ. A vector field Z on N is called a canonical vector field if it admits the representation Z = ∂/∂t + (H, ) where (,) denotes the Poisson bracket on M. For a given function σ, a prescription is given for the construction of the space c(σ)(N) of contact fields in terms of solutions F of the p.d.e. Z<F> = σh. The vector space (UNFORMATTED EQUATION FOLLOWS) c(N) = ∪ (σ∊C)(∞)c(σ)(N) (END UNFORMATTED EQUATION) is shown to have the structure of a Lie sub-algebra of the Lie algebra of vector fields on N. It is shown that the associated subspace A(π) = {X:X˩π = 0} is such that c(σ)(N) ∩ A(π) = {0}. This implies that there is an X in c(σ)(N) such that X˩π ≠ 0. Thus, if the function H that appears in the Cartan form π is such that H = X˩π for any X ∊ c(σ)(N) it is possible to deduce that ∂H/∂t ≠ 0, which shows that such vector fields may be of relevance to the theory of non-conservative systems. A different set of 2n 1-forms {π(h),πʰ} is introduced on N that are subject to analogous conditions which ensure the existence of local coordinates (qʰ,p(h)) on M and a function K(qʰ,p(h),t) that gives rise to a new Cartan form π on N such that dπ= π(h) ∧ πʰ. By definition, the fundamental invariant of a parameter-dependent canonical transformation on N is dπ = dπ. In this new setting a contact field X satisfies the ₤ₓπ = σπ for some function σ: N to R. The relationship between the contact vector fields X and X is investigated in depth.

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36

Schelp, Richard Charles. "The standard model and beyond in noncommutative geometry /." Digital version accessible at:, 2000. http://wwwlib.umi.com/cr/utexas/main.

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Dias, Ronaldo [UNESP]. "Terceiro problema de Hilbert e Teorema de Dehn." Universidade Estadual Paulista (UNESP), 2013. http://hdl.handle.net/11449/94274.

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Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-03-28Bitstream added on 2014-06-13T20:47:33Z : No. of bitstreams: 1 dias_r_me_sjrp.pdf: 272282 bytes, checksum: b1e44b8f25de4abb1780694b335cbf0c (MD5)<br>O objetivo principal deste trabalho é provar o Teorema de Dehn. Esse teorema é resposta ao Terceiro Problema de Hilbert, este problema refere-se à seguinte situação: Se dois poliedros possuem o mesmo volume eles são congruentes por corte, ou seja, é sempre possível tomar dois poliedros de mesmo volume e decompor um em poliedros menores de tal maneira que os reorganizando seja possível montar o outro. A resposta para esta questão é negativa e sua prova ficou conhecida como teorema de Dehn. Inicialmente estudaremos conceitos de área, volume e congruência por corte para figuras planas e no espaço. Nesta etapa discutiremos a decomposição de figuras em polígonos e poliedros. Em seguida usando algumas propriedades de funções aditivas e os ângulos diedros de um poliedro, construiremos um invariante que será a ferramenta principal na demonstração do Teorema de Dehn. Como considerações finais, cito o Paradoxo de Banach-Tarski, uma vez que o mesmo é relacionado naturalmente ao problema de congruência por corte e decomposição de figuras no espaço e apresento um capítulo com algumas atividades que podem ser desenvolvidas na educação básica<br>The main object of this work is study the Third Problem of Hilbert and the Dehn Theorem

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38

Dias, Ronaldo. "Terceiro problema de Hilbert e Teorema de Dehn /." São José do Rio Preto, 2013. http://hdl.handle.net/11449/94274.

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Orientador: Parham Salehyan<br>Banca: Ali Tahzibi<br>Banca: Luciana de Fátima Martins<br>O PROFMAT - Programa de Mestrado Profissional em Matemática em Rede Nacional é coordenado pela Sociedade Brasileira de Matemática e realizado por uma rede de Instituições de Ensino Superior.<br>Resumo: O objetivo principal deste trabalho é provar o Teorema de Dehn. Esse teorema é resposta ao Terceiro Problema de Hilbert, este problema refere-se à seguinte situação: Se dois poliedros possuem o mesmo volume eles são congruentes por corte, ou seja, é sempre possível tomar dois poliedros de mesmo volume e decompor um em poliedros menores de tal maneira que os reorganizando seja possível montar o outro. A resposta para esta questão é negativa e sua prova ficou conhecida como teorema de Dehn. Inicialmente estudaremos conceitos de área, volume e congruência por corte para figuras planas e no espaço. Nesta etapa discutiremos a decomposição de figuras em polígonos e poliedros. Em seguida usando algumas propriedades de funções aditivas e os ângulos diedros de um poliedro, construiremos um invariante que será a ferramenta principal na demonstração do Teorema de Dehn. Como considerações finais, cito o Paradoxo de Banach-Tarski, uma vez que o mesmo é relacionado naturalmente ao problema de congruência por corte e decomposição de figuras no espaço e apresento um capítulo com algumas atividades que podem ser desenvolvidas na educação básica<br>Abstract: The main object of this work is study the Third Problem of Hilbert and the Dehn Theorem<br>Mestre

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39

Baraglia, David. "G2 geometry and integrable systems." Thesis, University of Oxford, 2009. http://ora.ox.ac.uk/objects/uuid:30cf9c7c-157e-4204-b68b-08f6e199ef36.

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We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. We also relate various real forms of the Toda equations to minimal surfaces in quadrics of arbitrary signature. In the case of the Hitchin component for PSL(3,R) we provide a new proof of the relation to convex RP²-structures and hyperbolic affine spheres. For PSp(4,R) we prove such representations are the monodromy for a special class of projective structure on the unit tangent bundle of the surface. We prove these are isomorphic to the convex-foliated projective structures of Guichard and Wienhard. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, work which traces back to Cartan. Nurowski showed that there is an associated signature (2,3) conformal structure. We clarify this as a relationship between a parabolic geometry associated to the split real form of G₂ and a conformal geometry with holonomy in G₂. Moreover in terms of the conformal geometry we prove this distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. The moduli space of deformations of a compact coassociative submanifold L in a G₂ manifold is shown to have a natural local embedding as a submanifold of H2(L,R). We consider G2-manifolds with a T^4-action of isomorphisms such that the orbits are coassociative tori and prove a local equivalence to minimal 3-manifolds in R^{3,3} = H²(T⁴,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R^{3,3} and hence G₂-metrics from equations that are a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Ampere equation is explained.

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Nystrom, Michel. "The Ambrose-Palais-Singer theorem in synthetic differential geometry /." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=66260.

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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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Eastlick, Mark Thomas. "Discrete differential geometry and an application in multiresolution analysis." Thesis, University of Sheffield, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.434535.

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Brzezinski, Tomasz. "Differential geometry of quantum groups and quantum fibre bundles." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.321113.

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Jacobs, Andrew D. "Nonstandard quantum groups : twisting constructions and noncommutative differential geometry." Thesis, University of St Andrews, 1998. http://hdl.handle.net/10023/13693.

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The general subject of this thesis is quantum groups. The major original results are obtained in the particular areas of twisting constructions and noncommutative differential geometry. Chapters 1 and 2 are intended to explain to the reader what are quantum groups. They are written in the form of a series of linked results and definitions. Chapter 1 reviews the theory of Lie algebras and Lie groups, focusing attention in particular on the classical Lie algebras and groups. Though none of the quoted results are due to the author, such a review, aimed specifically at setting up the paradigm which provides essential guidance in the theory of quantum groups, does not seem to have appeared already. In Chapter 2 the elements of the quantum group theory are recalled. Once again, almost none of the results are due to the author, though in Section 2.10, some results concerning the nonstandard Jordanian group are presented, by way of a worked example, which have not been published. Chapter 3 concerns twisting constructions. We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as previously known examples of non-standard quantum groups. In particular we are able to construct generalisations of both the Cremmer-Gervais deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and Galindo in an explicit and straightforward manner. In Chapter 4 we consider the differential calculus on Hopf algebras as introduced by Woronowicz. We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL[sub]h,[sub]g(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL[sub]h(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL[sub]h,[sub]g(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL[sub]h(2). This 3-dimensional calculus may be obtained through a classical-like reduction from any one of the three families of 4-dimensional calculi on GL[sub]h,[sub]g(2). Details of the higher order calculi and also the quantum Lie algebras are presented for all calculi. The quantum Lie algebra obtained from the bicovariant calculus on SL[sub]h(2) is shown to be isomorphic to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian universal enveloping algebra U[sub]h(sl[sub]2(C)) and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module. We also obtain the quantum Killing form for this quantum Lie algebra.

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Athanasopoulos, Michael, Hassan Ugail, and Castro Gabriela Gonzalez. "Parametric design of aircraft geometry using partial differential equations." Elsevier, 2009. http://hdl.handle.net/10454/2725.

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Baker, Andrew. "Applications of differential geometry to high spin field theories." Thesis, Aston University, 1990. http://publications.aston.ac.uk/10645/.

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The main aim of this thesis is to investigate the application of methods of differential geometry to the constraint analysis of relativistic high spin field theories. As a starting point the coordinate dependent descriptions of the Lagrangian and Dirac-Bergmann constraint algorithms are reviewed for general second order systems. These two algorithms are then respectively employed to analyse the constraint structure of the massive spin-1 Proca field from the Lagrangian and Hamiltonian viewpoints. As an example of a coupled field theoretic system the constraint analysis of the massive Rarita-Schwinger spin-3/2 field coupled to an external electromagnetic field is then reviewed in terms of the coordinate dependent Dirac-Bergmann algorithm for first order systems. The standard Velo-Zwanziger and Johnson-Sudarshan inconsistencies that this coupled system seemingly suffers from are then discussed in light of this full constraint analysis and it is found that both these pathologies degenerate to a field-induced loss of degrees of freedom. A description of the geometrical version of the Dirac-Bergmann algorithm developed by Gotay, Nester and Hinds begins the geometrical examination of high spin field theories. This geometric constraint algorithm is then applied to the free Proca field and to two Proca field couplings; the first of which is the minimal coupling to an external electromagnetic field whilst the second is the coupling to an external symmetric tensor field. The onset of acausality in this latter coupled case is then considered in relation to the geometric constraint algorithm.

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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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Pearson, John Clifford. "The noncommutative geometry of ultrametric cantor sets." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24657.

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Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008.<br>Committee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ian

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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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StClair, Jessica Lindsey. "Geometry of Spaces of Planar Quadrilaterals." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/26887.

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The purpose of this dissertation is to investigate the geometry of spaces of planar quadrilaterals. The topology of moduli spaces of planar quadrilaterals (the set of all distinct planar quadrilaterals with fixed side lengths) has been well-studied [5], [8], [10]. The symplectic geometry of these spaces has been studied by Kapovich and Millson [6], but the Riemannian geometry of these spaces has not been thoroughly examined. We study paths in the moduli space and the pre-moduli space. We compare intraplanar paths between points in the moduli space to extraplanar paths between those same points. We give conditions on side lengths to guarantee that intraplanar motion is shorter between some points. Direct applications of this result could be applied to motion-planning of a robot arm. We show that horizontal lifts to the pre-moduli space of paths in the moduli space can exhibit holonomy. We determine exactly which collections of side lengths allow holonomy.<br>Ph. D.

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