Tesis: "Convergence of Riemannian manifolds"

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Zergänge, Norman [Verfasser]. "Convergence of Riemannian manifolds with critical curvature bounds / Norman Zergänge". Magdeburg : Universitätsbibliothek, 2017. http://d-nb.info/1141230488/34.

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Martins, Tiberio Bittencourt de Oliveira. "Newton's methods under the majorant principle on Riemannian manifolds". Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4847.

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Apresentamos, nesta tese, uma an álise da convergência do m étodo de Newton inexato com tolerância de erro residual relativa e uma an alise semi-local de m etodos de Newton robustos exato e inexato, objetivando encontrar uma singularidade de um campo de vetores diferenci avel de nido em uma variedade Riemanniana completa, baseados no princ pio majorante a m invariante. Sob hip oteses locais e considerando uma fun ção majorante geral, a Q-convergância linear do m etodo de Newton inexato com uma tolerância de erro residual relativa xa e provada. Na ausência dos erros, a an alise apresentada reobtem o teorema local cl assico sobre o m etodo de Newton no contexto Riemanniano. Na an alise semi-local dos m etodos exato e inexato de Newton apresentada, a cl assica condi ção de Lipschitz tamb em e relaxada usando uma fun ção majorante geral, permitindo estabelecer existência e unicidade local da solu ção, uni cando previamente resultados pertencentes ao m etodo de Newton. A an alise enfatiza a robustez, a saber, e dada uma bola prescrita em torno do ponto inicial que satifaz as hip oteses de Kantorovich, garantindo a convergência do m etodo para qualquer ponto inicial nesta bola. Al em disso, limitantes que dependem da função majorante para a taxa de convergência Q-quadr atica do m étodo exato e para a taxa de convergência Q-linear para o m etodo inexato são obtidos.
A local convergence analysis with relative residual error tolerance of inexact Newton method and a semi-local analysis of a robust exact and inexact Newton methods are presented in this thesis, objecting to nd a singularity of a di erentiable vector eld de ned on a complete Riemannian manifold, based on a ne invariant majorant principle. Considering local assumptions and a general majorant function, the Q-linear convergence of inexact Newton method with a xed relative residual error tolerance is proved. In the absence of errors, the analysis presented retrieves the classical local theorem on Newton's method in Riemannian context. In the semi-local analysis of exact and inexact Newton methods presented, the classical Lipschitz condition is also relaxed by using a general majorant function, allowing to establish the existence and also local uniqueness of the solution, unifying previous results pertaining Newton's method. The analysis emphasizes robustness, being more speci c, is given a prescribed ball around the point satisfying Kantorovich's assumptions, ensuring convergence of the method for any starting point in this ball. Furthermore, the bounds depending on the majorant function for Q-quadratic convergence rate of the exact method and Q-linear convergence rate of the inexact method are obtained.

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Luckhardt, Daniel [Verfasser], Thomas [Akademischer Betreuer] Schick, Thomas [Gutachter] Schick, Ralf [Gutachter] Meyer, Stephan [Gutachter] Huckemann, Russell [Gutachter] Luke, Viktor [Gutachter] Pidstrygach y Ingo [Gutachter] Witt. "Benjamini-Schramm Convergence of Normalized Characteristic Numbers of Riemannian Manifolds / Daniel Luckhardt ; Gutachter: Thomas Schick, Ralf Meyer, Stephan Huckemann, Russell Luke, Viktor Pidstrygach, Ingo Witt ; Betreuer: Thomas Schick". Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2020. http://d-nb.info/1209358239/34.

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Guevara, Stefan Alberto Gómez. "Unificando o análise local do método de Newton em variedades Riemannianas". Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/6951.

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In this work we consider the problem of finding a singularity of a field of differentiable vectors X on a Riemannian manifold. We present a local analysis of the convergence of Newton's method to find a singularity of field X on an increasing condition. The analysis shows a relationship between the major function and the vector field X. We also present a semi-local Kantorovich type analysis in the Riemannian context under a major condition. The two results allow to unify some previously unrelated results.
Neste trabalho consideramos o problema de encontrar uma singularidade de um campo de vetores diferenciável X sobre uma variedade Riemanniana. Apresentamos uma análise local da convergência do método de Newton para encontrar uma singularidade do Campo X sobre uma condição majorante. A análise mostra uma relação entre a função majorante e o campo de vetores X. Também apresentamos uma análise semi-local do tipo Kantorovich no contexto Riemanniana sob uma condição majorante. Os dois resultados permitem unificar alguns resultados não previamente.

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Erb, Wolfgang. "Uncertainty principles on Riemannian manifolds". kostenfrei, 2010. https://mediatum2.ub.tum.de/node?id=976465.

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6

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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7

Longa, Eduardo Rosinato. "Hypersurfaces of paralellisable Riemannian manifolds". reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2017. http://hdl.handle.net/10183/158755.

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Introduzimos uma aplicação de Gauss para hipersuperfícies de variedades Riemannianas paralelizáveis e definimos uma curvatura associada. Após, provamos um teorema de Gauss-Bonnet. Como exemplo, estudamos cuidadosamente o caso no qual o espaço ambiente é uma esfera Euclidiana menos um ponto e obtemos um teorema de rigidez topológica. Ele é utilizado para dar uma prova alternativa para um teorema de Qiaoling Wang and Changyu Xia, o qual afirma que se uma hipersuperfície orientável imersa na esfera está contida em um hemisfério aberto e tem curvatura de Gauss-Kronecker nãonula então ela é difeomorfa a uma esfera. Depois, obtemos alguns invariantes topol_ogicos para hipersuperfícies de variedades translacionais que dependem da geometria da variedade e do espaço ambiente. Finalmente, encontramos obstruções para a existência de certas folheações de codimensão um.
We introduce a Gauss map for hypersurfaces of paralellisable Riemannian manifolds and de ne an associated curvature. Next, we prove a Gauss- Bonnet theorem. As an example, we carefully study the case where the ambient space is an Euclidean sphere minus a point and obtain a topological rigidity theorem. We use it to provide an alternative proof for a theorem of Qiaoling Wang and Changyu Xia, which asserts that if an orientable immersed hypersurface of the sphere is contained in an open hemisphere and has nowhere zero Gauss-Kronecker curvature, then it is di eomorphic to a sphere. Later, we obtain some topological invariants for hypersurfaces of translational manifolds that depend on the geometry of the manifold and the ambient space. Finally, we nd obstructions to the existence of certain codimension-one foliations.

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Catalano, Domenico Antonino. "Concircular diffeomorphisms of pseudo-Riemannian manifolds /". [S.l.] : [s.n.], 1999. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=13064.

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Afsari, Bijan. "Means and averaging on riemannian manifolds". College Park, Md. : University of Maryland, 2009. http://hdl.handle.net/1903/9978.

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Thesis (Ph.D.) -- University of Maryland, College Park, 2009.
Thesis research directed by: Dept. of Mathematics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

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Popiel, Tomasz. "Geometrically-defined curves in Riemannian manifolds". University of Western Australia. School of Mathematics and Statistics, 2007. http://theses.library.uwa.edu.au/adt-WU2007.0119.

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[Truncated abstract] This thesis is concerned with geometrically-defined curves that can be used for interpolation in Riemannian or, more generally, semi-Riemannian manifolds. As in much of the existing literature on such curves, emphasis is placed on manifolds which are important in computer graphics and engineering applications, namely the unit 3-sphere S3 and the closely related rotation group SO(3), as well as other Lie groups and spheres of arbitrary dimension. All geometrically-defined curves investigated in the thesis are either higher order variational curves, namely critical points of cost functionals depending on (covariant) derivatives of order greater than 1, or defined by geometrical algorithms, namely generalisations to manifolds of algorithms from the field of computer aided geometric design. Such curves are needed, especially in the aforementioned applications, since interpolation methods based on applying techniques of classical approximation theory in coordinate charts often produce unnatural interpolants. However, mathematical properties of higher order variational curves and curves defined by geometrical algorithms are in need of substantial further investigation: higher order variational curves are solutions of complicated nonlinear differential equations whose properties are not well-understood; it is usually unclear how to impose endpoint derivative conditions on, or smoothly piece together, curves defined by geometrical algorithms. This thesis addresses these difficulties for several classes of curves. ... The geometrical algorithms investigated in this thesis are generalisations of the de Casteljau and Cox-de Boor algorithms, which define, respectively, polynomial B'ezier and piecewise-polynomial B-spline curves by dividing, in certain ratios and for a finite number of iterations, piecewise-linear control polygons corresponding to finite sequences of control points. We show how the control points of curves produced by the generalised de Casteljau algorithm in an (almost) arbitrary connected finite-dimensional Riemannian manifold M should be chosen in order to impose desired endpoint velocities and (covariant) accelerations and, thereby, piece the curves together in a C2 fashion. A special case of the latter construction simplifies when M is a symmetric space. For the generalised Cox-de Boor algorithm, we analyse in detail the failure of a fundamental property of B-spline curves, namely C2 continuity at (certain) knots, to carry over to M.

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Desa, Zul Kepli Bin Mohd. "Riemannian manifolds with Einstein-like metrics". Thesis, Durham University, 1985. http://etheses.dur.ac.uk/7571/.

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In this thesis, we investigate properties of manifolds with Riemannian metrics which satisfy conditions more general than those of Einstein metrics, including the latter as special cases. The Einstein condition is well known for being the Euler- Lagrange equation of a variational problem. There is not a great deal of difference between such metrics and metrics with Ricci tensor parallel for the latter are locally Riemannian products of the former. More general classes of metrics considered include Ricci- Codazzi and Ricci cyclic parallel. Both of these are of constant scalar curvature. Our study is divided into three parts. We begin with certain metrics in 4-dimensions and conclude our results with three theorems, the first of which is equivalent to a result of Kasner [Kal] while the second and part of the third is known to Derdzinski [Del.2].Next we construct the metrics mentioned above on spheres of odd dimension. The construction is similar to Jensen's [Jel] but more direct and is due essentially to Gray and Vanhecke [GV]. In this way we obtain .beside the standard metric, the second Einstein metric of Jensen. As for the Ricci- Codazzi metrics, they are essentially Einstein, but the Ricci cyclic parallel metrics seem to form a larger class. Finally, we consider subalgebras of the exceptional Lie algebra g2. Making use of computer programmes in 'reduce' we compute all the corresponding metrics on the quotient spaces associated with G2.

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Parmar, Vijay K. "Harmonic morphisms between semi-Riemannian manifolds". Thesis, University of Leeds, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.305696.

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Dahmani, Kamilia. "Weighted LP estimates on Riemannian manifolds". Thesis, Toulouse 3, 2018. http://www.theses.fr/2018TOU30188/document.

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Cette thèse s'inscrit dans le domaine de l'analyse harmonique et plus exactement, des estimations à poids. Un intérêt particulier est porté aux estimations Lp à poids des transformées de Riesz sur des variétés Riemanniennes complètes ainsi qu'à l'optimalité des résultats en terme de la puissance de la caractéristique des poids. On obtient un premier résultat (en terme de la linéarité et de la non dépendance de la dimension) sur des espaces pas nécessairement de type homogène, lorsque p = 2 et la courbure de Bakry-Emery est positive. On utilise pour cela une approche analytique en exhibant une fonction de Bellman concrète. Puis, en utilisant des techniques stochastiques et une domination éparse, on démontre que les transformées de Riesz sont bornées sur Lp, pour p ∈ (1, +∞) et on déduit également le résultat précèdent. Enfin, on utilise un changement élégant dans la preuve précèdente pour affaiblir l'hypothèse sur la courbure et la supposer minorée
The topics addressed in this thesis lie in the field of harmonic analysis and more pre- cisely, weighted inequalities. Our main interests are the weighted Lp-bounds of the Riesz transforms on complete Riemannian manifolds and the sharpness of the bounds in terms of the power of the characteristic of the weights. We first obtain a linear and dimensionless result on non necessarily homogeneous spaces, when p = 2 and the Bakry-Emery curvature is non-negative. We use here an analytical approach by exhibiting a concrete Bellman function. Next, using stochastic techniques and sparse domination, we prove that the Riesz transforms are Lp-bounded for p ∈ (1, +∞) and obtain the previous result for free. Finally, we use an elegant change in the precedent proof to weaken the condition on the curvature and assume it is bounded from below

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ZEESTRATEN, MARTINUS. "Programming by Demonstration on Riemannian Manifolds". Doctoral thesis, Università degli studi di Genova, 2018. http://hdl.handle.net/11567/930621.

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This thesis presents a Riemannian approach to Programming by Demonstration (PbD). It generalizes an existing PbD method from Euclidean manifolds to Riemannian manifolds. In this abstract, we review the objectives, methods and contributions of the presented approach. OBJECTIVES PbD aims at providing a user-friendly method for skill transfer between human and robot. It enables a user to teach a robot new tasks using few demonstrations. In order to surpass simple record-and-replay, methods for PbD need to ‘understand’ what to imitate; they need to extract the functional goals of a task from the demonstration data. This is typically achieved through the application of statisticalmethods. The variety of data encountered in robotics is large. Typical manipulation tasks involve position, orientation, stiffness, force and torque data. These data are not solely Euclidean. Instead, they originate from a variety of manifolds, curved spaces that are only locally Euclidean. Elementary operations, such as summation, are not defined on manifolds. Consequently, standard statistical methods are not well suited to analyze demonstration data that originate fromnon-Euclidean manifolds. In order to effectively extract what-to-imitate, methods for PbD should take into account the underlying geometry of the demonstration manifold; they should be geometry-aware. Successful task execution does not solely depend on the control of individual task variables. By controlling variables individually, a task might fail when one is perturbed and the others do not respond. Task execution also relies on couplings among task variables. These couplings describe functional relations which are often called synergies. In order to understand what-to-imitate, PbDmethods should be able to extract and encode synergies; they should be synergetic. In unstructured environments, it is unlikely that tasks are found in the same scenario twice. The circumstances under which a task is executed—the task context—are more likely to differ each time it is executed. Task context does not only vary during task execution, it also varies while learning and recognizing tasks. To be effective, a robot should be able to learn, recognize and synthesize skills in a variety of familiar and unfamiliar contexts; this can be achieved when its skill representation is context-adaptive. THE RIEMANNIAN APPROACH In this thesis, we present a skill representation that is geometry-aware, synergetic and context-adaptive. The presented method is probabilistic; it assumes that demonstrations are samples from an unknown probability distribution. This distribution is approximated using a Riemannian GaussianMixtureModel (GMM). Instead of using the ‘standard’ Euclidean Gaussian, we rely on the Riemannian Gaussian— a distribution akin the Gaussian, but defined on a Riemannian manifold. A Riev mannian manifold is a manifold—a curved space which is locally Euclidean—that provides a notion of distance. This notion is essential for statistical methods as such methods rely on a distance measure. Examples of Riemannian manifolds in robotics are: the Euclidean spacewhich is used for spatial data, forces or torques; the spherical manifolds, which can be used for orientation data defined as unit quaternions; and Symmetric Positive Definite (SPD) manifolds, which can be used to represent stiffness and manipulability. The Riemannian Gaussian is intrinsically geometry-aware. Its definition is based on the geometry of the manifold, and therefore takes into account the manifold curvature. In robotics, the manifold structure is often known beforehand. In the case of PbD, it follows from the structure of the demonstration data. Like the Gaussian distribution, the Riemannian Gaussian is defined by a mean and covariance. The covariance describes the variance and correlation among the state variables. These can be interpreted as local functional couplings among state variables: synergies. This makes the Riemannian Gaussian synergetic. Furthermore, information encoded in multiple Riemannian Gaussians can be fused using the Riemannian product of Gaussians. This feature allows us to construct a probabilistic context-adaptive task representation. CONTRIBUTIONS In particular, this thesis presents a generalization of existing methods of PbD, namely GMM-GMR and TP-GMM. This generalization involves the definition ofMaximum Likelihood Estimate (MLE), Gaussian conditioning and Gaussian product for the Riemannian Gaussian, and the definition of ExpectationMaximization (EM) and GaussianMixture Regression (GMR) for the Riemannian GMM. In this generalization, we contributed by proposing to use parallel transport for Gaussian conditioning. Furthermore, we presented a unified approach to solve the aforementioned operations using aGauss-Newton algorithm. We demonstrated how synergies, encoded in a Riemannian Gaussian, can be transformed into synergetic control policies using standard methods for LinearQuadratic Regulator (LQR). This is achieved by formulating the LQR problem in a (Euclidean) tangent space of the Riemannian manifold. Finally, we demonstrated how the contextadaptive Task-Parameterized Gaussian Mixture Model (TP-GMM) can be used for context inference—the ability to extract context from demonstration data of known tasks. Our approach is the first attempt of context inference in the light of TP-GMM. Although effective, we showed that it requires further improvements in terms of speed and reliability. The efficacy of the Riemannian approach is demonstrated in a variety of scenarios. In shared control, the Riemannian Gaussian is used to represent control intentions of a human operator and an assistive system. Doing so, the properties of the Gaussian can be employed to mix their control intentions. This yields shared-control systems that continuously re-evaluate and assign control authority based on input confidence. The context-adaptive TP-GMMis demonstrated in a Pick & Place task with changing pick and place locations, a box-taping task with changing box sizes, and a trajectory tracking task typically found in industry

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15

Finkelstein, Shlomit Ritz. "Gravity in hyperspin manifolds". Diss., Georgia Institute of Technology, 1987. http://hdl.handle.net/1853/27974.

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Lu, Zhuoran. "Properties of Soft Maps on Riemannian Manifolds". Thesis, New York University, 2017. http://pqdtopen.proquest.com/#viewpdf?dispub=10617234.

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This paper concerns the soft map f from a Riemannian manifold to a probability space that minimizes the Dirichlet energy. First we give the explicit formula from any Riemannian manifold M to Prob(R). Secondly we discuss the map from M to Prob(R^d), prove the classic boundary condition implies classic solution. Then we proceed to the map from M to Prob(N), where N is a Riemannian manifold, and shows that if N is non-positive curvature, simply-connected, f has classic boundary condition, then f is classic solution and a harmonic map. Counter-examples are given when some of the above conditions are not fulfilled. In the last part we restrict the discussion in Gaussian measures. Using the Riemannian structure of the space of Gaussian measures, we prove an old result with a new method. We also show the soft map from M to non-degenerate Gaussian measures on R^d is harmonic map, give the explicit formula for the soft map in a special case.

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Kangaslampi, Riikka. "Uniformly quasiregular mappings on elliptic riemannian manifolds /". Helsinki : Suomalainen Tiedeakat, 2008. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=018603114&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.

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Friswell, Robert Michael. "Harmonic vector fields on pseudo-Riemannian manifolds". Thesis, University of York, 2014. http://etheses.whiterose.ac.uk/7878/.

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This thesis generalises the theory of harmonic vector fields to the non-compact pseudo- Riemannian case. After introducing the required background theory we consider the first variation of the local energies to find the Euler-Lagrange equations for this new case. We then introduce a natural closed conformal gradient field on pseudo-Riemannian warped products and find the Euler-Lagrange equations for harmonic closed conformal vector fields of this sort. We then give examples of such harmonic closed conformal fields, this leads to a harmonic vector fields on a 2-sphere with a rotationally symmetric singular metric. The harmonic conformal gradient fields on all hyperquadrics are then categorised up to con- gruence. The harmonic Killing fields on the 2-dimensional hyperquadrics are found, and shown to be unique up to congruence.

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19

Garcia-Leon, Joel. "Cheeger constant and isoperimetric inequalities on Riemannian manifolds". Thesis, Imperial College London, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.417041.

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Alcantara, Priscila Rodrigues de. "Hypersurfaces with prescribed mean curvature in Riemannian manifolds". Universidade Federal do CearÃ, 2010. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=5278.

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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
This work shows results existence and uniqueness of graphs with prescribed mean curvature. We demonstrate that a natural fixation Dirichlet problem for graphs of average curvature is required to consider those graphs like leaves on a Riemannian submersion Killing transversal cylinder, the cylinder given by flow lines of a Killing vector field. Using this approach, we are able to solve the problem in a way more comprehensive, giving a unified proof and existence results.
O objetivo deste trabalho Ã exibir resultados de existÃncia e unicidade de grÃficos com curvatura mÃdia prescrita. Demonstraremos que uma fixacÃoÂ natural do problema de Dirichlet para grÃficos de curvatura mÃdia prescrita Ã considerar esses grÃficos como folhas em uma submersÃo Riemanniana transversal ao cilindro de Killing, isto Ã, ao cilindro dado pelas linhasde fluxo de um campo de vetores de Killing. Usando essa aproximaÃÃo, somos capazes de resolver o problema em um modo mais compreensivo, dando uma prova unificada e resultados de existÃncia para uma ampla gama do ambiente de variedades Riemannianas.

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Barrett, Dennis Ian. "Contributions to the study of nonholonomic Riemannian manifolds". Thesis, Rhodes University, 2017. http://hdl.handle.net/10962/7554.

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Menegaz, Henrique Marra Taira. "Unscented kalman filtering on euclidean and riemannian manifolds". reponame:Repositório Institucional da UnB, 2016. http://repositorio.unb.br/handle/10482/21617.

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Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2016.
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Nesta tese, nós estudamos com profundidade uma técnica cada vez mais popular conhecida como Filtro de Kalman Unscented (FKU). Consideremos tanto aspectos teóricos como práticos da filtragem Unscented. Na primeira parte deste trabalho, propomos uma sistematização da teoria de filtragem de Kalman Unscented. Nessa sistematização nós i) agrupamos todos os FKUs da literatura, ii) apresentamos correções para inconsistências teóricas detectadas, e iii) propomos uma ferramenta para a construção de novos FKU's de forma consistente. Essencialmente, essa sistematização é feita mediante a revisão dos conceitos de conjunto sigma (SS), Transformação Unscented (TU), Transformação Unscented Escalada (TUE), Transformação Unscented Raiz-Quadrada (TURQ), FKU, e Filtro de Kalman Unscented Raiz-Quadrada (FKURQ). Introduzimos FKUs tempo-contínuo e tempo-contínuo-discreto. Ilustramos os resultados em i) alguns exemplos analíticos e numéricos, e ii) um experimento prático que consiste em estimar a posição de uma válvula de aceleração eletrônica utilizando FKUs desenvolvidos neste trabalho; essa estimação da posição de válvula é também uma contribuição por si só desde um ponto de vista tecnológico. Na segunda parte, primeiro, nós i) revelamos inconsistência na teoria por trás dos FKUs e FKURQs para sistemas de quatérnios unitários da literatura — tais como definições de quatérnios aleatórios e de sistemas quaterniônicos com ruídos aditivos —, ii) propomos um FKU englobando todos esses FKU's, e iii) propomos um FKURQ com propriedades numéricas superiores a esses FKURQs. Segundo, propomos uma extensão de alguns resultados da literatura relativos a estatísticas em variedades Riemannianas. Terceiro, usamos esses resultados estatísticos para apresentar uma extensão para sistemas riemannianos da sistematização euclidiana desenvolvida na primeira parte. Nessa sistematização riemanniana, introduzimos i) sistemas riemannianos com ruídos aditivos; e versões riemannianas dos conceitos de SS, TU, TUE, TURQ, FKU, e FKURQ. Diversos novos FKUs são introduzidos. Depois, apresentamos formas fechadas para quase todas as operações contidas nos filtros riemannianos para sistemas de quatérnios unitários. Também introduzimos consistentes i) FKUs para sistemas de quatérnios unitários duais, e ii) FKUs tempo-contínuo e tempo-contínuo-discreto. __________________________________________________________________________________________________ ABSTRACT
In this thesis, we take an in-depth study of an increasingly popular estimation technique known as Unscented Kalman Filter (UKF). We consider theoretical and practical aspects of the unscented filtering. In the first part of this work, we propose a systematization of the Unscented Kalman filtering theory on Euclidean spaces. In this systematization, we i) gather all available UKF's in the literature, ii) present corrections to theoretical inconsistencies, and iii) provide a tool for the construction of new UKF's in a consistent way. Mainly, this systematization is done by revisiting the concepts of sigma set (SS), Unscented Transformation (UT), Scaled Unscented Transformation (SUT), Square-Root Unscented Transformation (SRUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). We introduce continuous-time and continuous-discrete-time UKF's. We illustrate the results in i) some analytical and numerical examples, and ii) a practical experiment consisting of estimating the position of an automotive electronic throttle valve using UKF's developed in this work; this valve's position estimation is also, from a technological perspective, a contribution on its own. In the second part, first, we i) unfold some consistence issues in the theory behind the UKF's and SRUKF's for unit quaternion systems of the literature—such as definitions of random quaternions and additive-noise quaternion systems—, ii) propose an UKF embodying all these UKF's, and iii) propose an SRUKF with better computational properties than all these SRUKF's. Second, we propose an extension of some results of the literature concerning statistics on Riemannian manifolds. Third, we use these statistical results to present an extension to Riemannian systems of the Euclidean systematization developed in the first part. In this Riemannian systematization, we propose i) additive-noise Riemannian systems; and ii) Riemannian versions of the concepts of SS, UT, SUT, SRUT, UKF, and SRUKF. Several new consistent UKF's are introduced. Afterwards, we present closed forms of almost all the operations contained in the Unscented-type Riemannian filters for unit quaternion systems. We also introduce consistent i) UKF's for systems of unit dual quaternions, and ii) continuous-time and continuous-discrete-time UKF's for Riemannian manifolds.

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23

VALTORTA, DANIELE. "ON THE P-LAPLACE OPERATOR ON RIEMANNIAN MANIFOLDS". Doctoral thesis, Università degli Studi di Milano, 2013. http://hdl.handle.net/2434/217559.

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This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.

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24

TENCONI, MARINA. "Localization for Riesz Means on compact Riemannian manifolds". Doctoral thesis, Università degli Studi di Milano-Bicocca, 2016. http://hdl.handle.net/10281/101979.

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This thesis deals with the problem of localization for the Riesz means for eigenfunction expansions of the Laplace-Beltrami operator. The classical Riemann localization principle states that if an integrable function of one variable vanishes in an open set, then its trigonometric Fourier expansion converges to zero in this set. This localization principle fails in R^d with d>=2. In order to recover localization one has to use suitable summability methods, such as the Bochner-Riesz means. In the first chapter of the thesis we focus on the compact rank one symmetric spaces case. While in the second chapter we show how some of the results obtained in the first chapter can be generalize to smooth compact and connected Riemannian manifolds. Consider a d-dimensional compact rank one symmetric space. To every square integrable function, and more generally tempered distribution, one can associate a Fourier series, i.e. an eigenfunction expansion of the Laplace-Beltrami operator. These Fourier series converge in the metric of L^2 and in the topology of distributions, but in general one cannot ensure the pointwise convergence. For this reason we introduce the summability method of Bochner-Riesz means: S_R^\alpha f(x). When \alpha=0 one obtains the spherical partial sums, which are a natural analogue of the partial sums of one-dimensional Fourier series in the Euclidean space. There are examples of the failure of localization in Holder, Lebesgue and Sobolev spaces. Despite the negative results, it has been proved by A.I. Basatis and C. Meaney that there is an almost everywhere localization principle for square integrable functions on compact rank one symmetric spaces; that is, if a square integrable function vanishes almost everywhere in an open set, then its Fourier series is equal to zero for almost every point in this open set. On the other hand, it is known that for square integrable functions localization holds everywhere above the so-called critical index \alpha=(d-1)/2, while for integrable functions the critical index is \alpha=d-1. In this work we continue this line of research in the area of exceptional sets in harmonic analysis. In particular we prove that for Bochner-Riesz means of order \alpha of p-integrable functions on compact rank one symmetric spaces localization holds, with a possible exception in a set of point of suitable Hausdorff dimension. More generally we consider localization for distributions in Sobolev spaces. Some of the result on compact rank one symmetric spaces can be generalize to a general smooth, connected and compact Riemannian manifold without boundary.

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25

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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26

Peng, Xiao-Wei. "Kollaps Riemannscher Mannigfaltigkeiten". Bonn : [s.n.], 1988. http://catalog.hathitrust.org/api/volumes/oclc/18440158.html.

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27

Taringoo, Farzin. "Control and optimization of hybrid systems on Riemannian manifolds". Thesis, McGill University, 2013. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=114351.

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The fundamental motivation for the work in this thesis is the analysis of the optimal control of hybrid systems on Riemannian manifolds using the language of differential geometry. Hybrid systems theory constitutes one of the major frameworks within which one may model and analyze the behaviour of large and complex systems; in particular, the optimal control of hybrid systems has been a focus of research over the last decades resulting in the important generalization of Minimum (Maximum) Principle of classic optimal control to hybrid systems. In the work of Shaikh and Caines (2007) and their predecessors, a formulation for a class of optimal control problems for general hybrid systems with nonlinear dynamics and autonomous or controlled switchings at switching states and times is proposed. In this thesis we extend the framework of Shaikh and Caines (2007) to a general class of hybrid systems defined on Riemannian manifolds. Due to the formulation generality, this class of hybrid systems covers a vast range of practical examples arising in such different areas as mechanical systems, chemical processes, air traffic control systems and cooperative robotic manipulator systems. In this thesis, a formulation for general hybrid systems on differentiable Riemannian manifolds is first presented. In the case of autonomous switchings, switching manifolds are modelled by embedded orientable submanifolds of the ambient state manifold and consequently hybrid optimal control problems are defined for hybrid systems in this general setting. Second, the classic Minimum Principle is extended to the Hybrid Minimum Principle (HMP) yielding the optimality necessary conditions for hybrid systems at the optimal switching states and times. The HMP statement in this thesis is obtained by employing the so-called needle control variation in the control value space. This class of control variations results in state trajectory variations along the nominal state trajectory in the ambient state manifold where the optimality conditions are derived by analyzing the cost function variation with respect to state variations. Third, in order to optimize switching states and times, numerical optimization algorithms (Gradient Geodesic-HMP, Newton Geodesic-HMP) are formulated by employing the HMP equations on general Riemannian state manifolds. The convergence analysis for the proposed algorithms is based upon the LaSalle Invariance Theorem. Technically these algorithms generalize the standard steepest descent and Newton methods in Euclidean spaces to Reimannian manifolds by employing the notion of Levi-Civita connections. Fourth, the derivation of the HMP results for hybrid systems on Riemannian manifolds is carried out for hybrid systems on Lie groups. The group structure of the ambient state manifold gives rise to a special form for the adjoint processes and Hamiltonian functions as the solutions for the optimality equations. In this thesis hybrid optimal control problems on Lie groups are only considered for the class of left invariant systems, however, the analysis can be easily modified to right invariant systems. In the setting of left invariant hybrid systems on Lie groups, the Gradient Geodesic-HMP and Newton Geodesic-HMP algorithm are modified into algorithms called the Gradient Exponential-HMP and Newton Exponential-HMP algorithms. The fifth and last part of the thesis focuses on the problem of optimization of autonomous hybrid optimal control problems with respect to the geometrical features of switching manifolds. Such features include first order and second order information on the switching manifolds such as curvature tensors and normal differential forms.
La motivation première du travail accompli dans cette thèse est l'analyse du contrôle optimal de systèmes hybrides sur les variétés riemanniennes en utilisant le language de la géométrie differentielle. La théorie des systèmes hybrides constitue un des cadres majeurs dans lequel on peut modeler et analyser le comportement de systèmes grands et complexes; en particulier, le contrôle optimal de systèmes hybrides a été le centre d'intérêt des recherches dans les décennies précédentes ayant comme résultat une importante généralisation du Principe Minimum (Maximum) du contrôle optimal classique aux systèmes hybrides.Le travail de Shaikh et Caines (2007) et leurs prédécesseurs propose une formule pour une classe de problèmes de contrôle optimal pour les systèmes hybrides généraux avec des dynamiques non linéaires et autonomes ou des commutations contrôlées aux états et temps de commutation. Cette thèse élargit le cadre de Shaikh et Caines (2007) à une classe générale de systèmes hybrides définis sur les variétés riemanniennes. En raison de la nature générale de la formulation, cette classe de systèmes hybrides couvre un vaste éventail d'exemples pratiques survenant dans différents domaines tels que les systèmes mécaniques, les procédés chimiques, le contrôle des systèmes de navigation aérienne, ainsi que les systèmes de manipulation de la robotique coopérative. Premièrement, cette thèse présente une formulation pour le cas des systèmes hybrides généraux sur les variétés riemaniennes différentielles. Dans le cas des commutations autonomes, les variétés de commutation sont modélisées par les sous-variétés prolongées et orientables de la variété d'état ambiante et conséquemment, les problèmes de contrôle optimal hybrides sont définis pour les systèmes hybrides dans ce contexte général. Deuxièmement, le Principe Minimum classique est étendu au Principe Minimum Hybride (HMP), produisant les conditions nécessaires d'optimalité pour les systèmes hybrides aux états et temps optimaux de commutation. L'énoncé du Principe Minimum Hybride (HMP) dans cette thèse est obtenu en utilisant la commande de variation d'aiguille, ainsi nommée, dans l'espace de valeur de contrôle. Cette classe de variation donne des variations de trajectoire au long de la trajectoire d'état nominale dans la variété d'état ambiante. Les conditions d'optimalité sont obtenues en analysant la variation de la fonction de coût en respectant les variations d'état. Finalement, la dernière partie de la thèse met l'accent sur la question d'optimisation des problèmes de contrôle optimal autonomes hybrides en ce qui concerne les caractéristiques géométriques des variétés de commutation. De telles caractéristiques comprennent des informations de premier et de second ordre sur les variétés de commutation telles que les tenseurs de courbures et les formes différentielles normales.

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28

CHIRA, JOSE LUIS LIZARBE. "ASYMPTOTIC LINKING INVARIANTS FOR RKACTIONS IN COMPACT RIEMANNIAN MANIFOLDS". PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2005. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=7761@1.

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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Arnold no seu trabalho The asymptotic Hopf Invariant and its applications de 1986, considerou sobre um domínio (ômega maiúsculo) compacto de R3 com bordo suave e homología trivial campos X e Y de divergência nula e tangentes ao bordo de (ômega maiúsculo) e definiu o índice de enlaçamento assintótico lk(X; Y ) e o invariante de Hopf associados a X e Y pela integral I(X; Y ) = (integral em ômega maiúsculo de alfa produto d- beta), onde (d-alfa) = iX-vol e (d-beta) = iy-vol, e mostrou que I(X; Y ) = lk(X; Y ). Agora, no presente trabalho estenderemos estas definições de índices de enlaçamento assintótico lk(fi maiúsculo,xi maiúsculo) e de invariante de Hopf I(fi maiúsculo,xi maiúsculo), onde (fi maiúsculo) e (xi maiúsculo) são ações de Rk e de Rs, k+s = n-1, respectivamente de difeomorfismos que preservam volume em (ômega maiúsculo n) a bola unitária fechada em Rn e mostraremos que lk (fi maiúsculo, xi maiúsculo) = I(fi maiúsculo,xi maiúsculo).
V.I. Arnold, in his paper The algebraic Hopf invariant and its applications published in 1986, considered a compact domain (ômega maiúsculo) in R3 with a smooth boundary and trivial homology and two divergence free vector fields X and Y in (ômega maiúsculo) tangent to the boundary. He defined an asymptotic linking invariant lk(X; Y ) and a Hopf invariant associated to X and Y by the integral I(X; Y ) = (integral em ômega maiúsculo de alfa produto d-beta) where (d-alfa) = iX-vol e (d-beta) = iy- vol. He showed that que I(X; Y ) = lk(X; Y ). In the present work we extend these definitions of the asymptotic linking invariant lk(fi maiúsculo,xi maiúsculo) and the Hopf invariant I(fi maiúsculo,xi maiúsculo) where (fi maiúsculo) and (xi maiúsculo) are actions Rk and Rs, k+s = n-1 by volume preserving diffeomorphisms, on the closed unit ball (ômega maiúsculo n) in and we show lk (fi maiúsculo, xi maiúsculo) = I(fi maiúsculo,xi maiúsculo).

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29

Sathaye, Bakul Sathaye. "Obstructions to Riemannian smoothings of locally CAT(0) manifolds". The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1531416481628579.

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30

Ndumu, Martin Ngu. "Brownian motion and the heat kernel on Riemannian manifolds". Thesis, University of Warwick, 1989. http://wrap.warwick.ac.uk/108565/.

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31

Park, Jiewon. "Convergence of complete Ricci-βat manifolds". Thesis, Massachusetts Institute of Technology, 2020. https://hdl.handle.net/1721.1/126935.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 57-59).
This thesis is focused on the convergence at inαnity of complete Ricci βat manifolds. In the αrst part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at inαnity has smooth cross section. The identiαcation map is given as the gradient βow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric βows.
by Jiewon Park.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics

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32

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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33

Osipova, Daria. "Symmetric submanifolds in symmetric spaces". Thesis, University of Hull, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.342976.

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34

D'Azevedo, Breda A. M. R. "Isometric foldings". Thesis, University of Southampton, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.235197.

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35

Bär, Christian y Frank Pfäffle. "Wiener measures on Riemannian manifolds and the Feynman-Kac formula". Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/5999/.

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This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.

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36

Emmerich, Patrick [Verfasser]. "Rigidity of Complete Riemannian Manifolds without Conjugate Points / Patrick Emmerich". Aachen : Shaker, 2013. http://d-nb.info/1049384369/34.

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37

Miker, Julie. "Eigenvalue Inequalities for a Family of Spherically Symmetric Riemannian Manifolds". UKnowledge, 2009. http://uknowledge.uky.edu/gradschool_diss/783.

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This thesis considers two isoperimetric inequalities for the eigenvalues of the Laplacian on a family of spherically symmetric Riemannian manifolds. The Payne-Pólya-Weinberger Conjecture (PPW) states that for a bounded domain Ω in Euclidean space Rn, the ratio λ1(Ω)/λ0(Ω) of the first two eigenvalues of the Dirichlet Laplacian is bounded by the corresponding eigenvalue ratio for the Dirichlet Laplacian on the ball BΩof equal volume. The Szegö-Weinberger inequality states that for a bounded domain Ω in Euclidean space Rn, the first nonzero eigenvalue of the Neumann Laplacian μ1(Ω) is maximized on the ball BΩ of the same volume. In the first three chapters we will look at the known work for the manifolds Rn and Hn. Then we will take a family a spherically symmetric manifolds given by Rn with a spherically symmetric metric determined by a radially symmetric function f. We will then give a PPW-type upper bound for the eigenvalue gap, λ1(Ω) − λ0(Ω), and the ratio λ1(Ω)/λ0(Ω) on a family of symmetric bounded domains in this space. Finally, we prove the Szegö-Weinberger inequality for this same class of domains.

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38

Whiting, James K. (James Kalani) 1980. "Path optimization using sub-Riemannian manifolds with applications to astrodynamics". Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/63035.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 131).
Differential geometry provides mechanisms for finding shortest paths in metric spaces. This work describes a procedure for creating a metric space from a path optimization problem description so that the formalism of differential geometry can be applied to find the optimal paths. Most path optimization problems will generate a sub-Riemannian manifold. This work describes an algorithm which approximates a sub-Riemannian manifold as a Riemannian manifold using a penalty metric so that Riemannian geodesic solvers can be used to find the solutions to the path optimization problem. This new method for solving path optimization problems shows promise to be faster than other methods, in part because it can easily run on parallel processing units. It also provides some geometrical insights into path optimization problems which could provide a new way to categorize path optimization problems. Some simple path optimization problems are described to provide an understandable example of how the method works and an application to astrodynamics is also given.
by James K. Whiting.
Ph.D.

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39

Stenzel, Matthew B. (Matthew Briggs). "Kähler structures on cotangent bundles of real analytic Riemannian manifolds". Thesis, Massachusetts Institute of Technology, 1990. http://hdl.handle.net/1721.1/49577.

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40

Ream, Robert. "Darboux Intergrability Of Wave Maps Into 2-Dimensional Riemannian Manifolds". DigitalCommons@USU, 2008. https://digitalcommons.usu.edu/etd/203.

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The harmonic map equations can be represented geometrically as an exterior differential system (EDS), E. Using this representation we study the harmonic maps from 2D Minkowski space into 2D Riemannian manifolds. These are also known as wave maps. In this case, E is invariant under conformal transformations of Minkowski space. The quotient of E by these conformal transformations, E/G, is an s=0 hyperbolic system. The main result of our study is that the prolonged EDS, E(k), is Darboux integrable if and only if the prolonged quotient EDS, E/G(k+1), is Darboux integrable. We also find invariants determining the Darboux integrability of both systems. Analyzing these invariants leads to three additional results. First, Darboux integrability of E, without prolongation, requires that the range manifold have zero scalar curvature. Second, after one prolongation there are two inequivalent metrics for which E(1) /super> is Darboux integrable. Third, prolonging to E(2) does not provide any further metrics with Darboux integrable wave maps.

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41

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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42

Raske, David Timothy. "Q-curvature on closed Riemannian manifolds of dimension greater than four /". Diss., Digital Dissertations Database. Restricted to UC campuses, 2005. http://uclibs.org/PID/11984.

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43

Franke, Dirk Christoph. "Quasiregular mappings and Hölder continuity of differential forms on Riemannian manifolds". [S.l. : s.n.], 1999. http://www.diss.fu-berlin.de/1999/57/index.html.

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44

Fonn, Eivind. "Computing Metrics on Riemannian Shape Manifolds : Geometric shape analysis made practical". Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2009. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9868.

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Shape analysis and recognition is a field ripe with creative solutions and innovative algorithms. We give a quick introduction to several different approaches, before basing our work on a representation introduced by Klassen et. al., considering shapes as equivalence classes of closed curves in the plane under reparametrization, and invariant under translation, rotation and scaling. We extend this to a definition for nonclosed curves, and prove a number of results, mostly concerning under which conditions on these curves the set of shapes become manifolds. We then motivate the study of geodesics on these manifolds as a means to compute a shape metric, and present two methods for computing such geodesics: the shooting method from Klassen et. al. and the ``direct'' method, new to this paper. Some numerical experiments are performed, which indicate that the direct method performs better for realistically chosen parameters, albeit not asymptotically.

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45

Tashiro, Kenshiro. "Gromov-Hausdorff limits of compact Heisenberg manifolds with sub-Riemannian metrics". Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263433.

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46

Zuddas, Daniele. "Branched coverings and 4-manifolds". Doctoral thesis, Scuola Normale Superiore, 2007. http://hdl.handle.net/11384/85677.

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47

Schüth, Dorothee. "Stetige isospektrale Deformationen". Bonn : [s.n.], 1994. http://catalog.hathitrust.org/api/volumes/oclc/31760957.html.

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48

Renesse, Max-K. von. "Comparison properties of diffusion semigroups on spaces with lower curvature bounds". Bonn : Mathematisches Institut der Universität Bonn, 2003. http://catalog.hathitrust.org/api/volumes/oclc/52348149.html.

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49

von, Deylen Stefan Wilhelm [Verfasser]. "Numerical Approximation in Riemannian Manifolds by Karcher Means / Stefan Wilhelm von Deylen". Berlin : Freie Universität Berlin, 2015. http://d-nb.info/1066645108/34.

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50

Weber, Patrick. "Cohomology groups on hypercomplex manifolds and Seiberg-Witten equations on Riemannian foliations". Doctoral thesis, Universite Libre de Bruxelles, 2017. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/252914.

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The thesis comprises two parts. In the first part, we investigate various cohomological aspects of hypercomplex manifolds and analyse the existence of special metrics. In the second part, we define Seiberg-Witten equations on the leaf space of manifolds which admit a Riemannian foliation of codimension four.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

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Tesis sobre el tema "Convergence of Riemannian manifolds"

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