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1
Farina, Sofia. "Barycentric Subspace Analysis on the Sphere and Image Manifolds." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/15797/.
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In this dissertation we present a generalization of Principal Component Analysis (PCA) to Riemannian manifolds called Barycentric Subspace Analysis and show some applications. The notion of barycentric subspaces has been first introduced first by X. Pennec. Since they lead to hierarchy of properly embedded linear subspaces of increasing dimension, they define a generalization of PCA on manifolds called Barycentric Subspace Analysis (BSA). We present a detailed study of the method on the sphere since it can be considered as the finite dimensional projection of a set of probability densities that have many practical applications. We also show an application of the barycentric subspace method for the study of cardiac motion in the problem of image registration, following the work of M.M. Rohé.
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Lord, Steven. "Riemannian non-commutative geometry /." Title page, abstract and table of contents only, 2002. http://web4.library.adelaide.edu.au/theses/09PH/09phl8661.pdf.
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Maignant, Elodie. "Plongements barycentriques pour l'apprentissage géométrique de variétés : application aux formes et graphes." Electronic Thesis or Diss., Université Côte d'Azur, 2023. http://www.theses.fr/2023COAZ4096.
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Une image obtenue par IRM, c'est plus de 60 000 pixels. La plus grosse protéine connue chez l'être humain est constituée d'environ 30 000 acides aminés. On parle de données en grande dimension. En réalité, la plupart des données en grande dimension ne le sont qu'en apparence. Par exemple, de toutes les images que l'on pourrait générer aléatoirement en coloriant 256 x 256 pixels, seule une infime proportion ressemblerait à l'image IRM d'un cerveau humain. C'est ce qu'on appelle la dimension intrinsèque des données. En grande dimension, apprentissage rime donc souvent avec réduction de dimension. Il existe de nombreuses méthodes de réduction de dimension, les plus récentes pouvant être classées selon deux approches.Une première approche, connue sous le nom d'apprentissage de variétés (manifold learning) ou réduction de dimension non linéaire, part du constat que certaines lois physiques derrière les données que l'on observe ne sont pas linéaires. Ainsi, espérer expliquer la dimension intrinsèque des données par un modèle linéaire est donc parfois irréaliste. Au lieu de cela, les méthodes qui relèvent du manifold learning supposent un modèle localement linéaire.D'autre part, avec l'émergence du domaine de l'analyse statistique de formes, il y eu une prise de conscience que de nombreuses données sont naturellement invariantes à certaines symétries (rotations, permutations, reparamétrisations...), invariances qui se reflètent directement sur la dimension intrinsèque des données. Ces invariances, la géométrie euclidienne ne peut pas les retranscrire fidèlement. Ainsi, on observe un intérêt croissant pour la modélisation des données par des structures plus fines telles que les variétés riemanniennes. Une deuxième approche en réduction de dimension consiste donc à généraliser les méthodes existantes à des données à valeurs dans des espaces non-euclidiens. On parle alors d'apprentissage géométrique. Jusqu'à présent, la plupart des travaux en apprentissage géométrique se sont focalisés sur l'analyse en composantes principales.Dans la perspective de proposer une approche qui combine à la fois apprentissage géométrique et manifold learning, nous nous sommes intéressés à la méthode appelée locally linear embedding, qui a la particularité de reposer sur la notion de barycentre, notion a priori définie dans les espaces euclidiens mais qui se généralise aux variétés riemanniennes. C'est d'ailleurs sur cette même notion que repose une autre méthode appelée barycentric subspace analysis, et qui fait justement partie des méthodes qui généralisent l'analyse en composantes principales aux variétés riemanniennes. Ici, nous introduisons la notion nouvelle de plongement barycentrique, qui regroupe les deux méthodes. Essentiellement, cette notion englobe un ensemble de méthodes dont la structure rappelle celle des méthodes de réduction de dimension linéaires et non linéaires, mais où le modèle (localement) linéaire est remplacé par un modèle barycentrique -- affine.Le cœur de notre travail consiste en l'analyse de ces méthodes, tant sur le plan théorique que pratique. Du côté des applications, nous nous intéressons à deux exemples importants en apprentissage géométrique : les formes et les graphes. En particulier, on démontre que par rapport aux méthodes standard de réduction de dimension en analyse statistique des graphes, les plongements barycentriques se distinguent par leur meilleure interprétabilité. En plus des questions pratiques liées à l'implémentation, chacun de ces exemples soulève ses propres questions théoriques, principalement autour de la géométrie des espaces quotients. Parallèlement, nous nous attachons à caractériser géométriquement les plongements localement barycentriques, qui généralisent la projection calculée par locally linear embedding. Enfin, de nouveaux algorithmes d'apprentissage géométrique, novateurs dans leur approche, complètent ce travailAn MRI image has over 60,000 pixels. The largest known human protein consists of around 30,000 amino acids. We call such data high-dimensional. In practice, most high-dimensional data is high-dimensional only artificially. For example, of all the images that could be randomly generated by coloring 256 x 256 pixels, only a very small subset would resemble an MRI image of a human brain. This is known as the intrinsic dimension of such data. Therefore, learning high-dimensional data is often synonymous with dimensionality reduction. There are numerous methods for reducing the dimension of a dataset, the most recent of which can be classified according to two approaches.A first approach known as manifold learning or non-linear dimensionality reduction is based on the observation that some of the physical laws behind the data we observe are non-linear. In this case, trying to explain the intrinsic dimension of a dataset with a linear model is sometimes unrealistic. Instead, manifold learning methods assume a locally linear model.Moreover, with the emergence of statistical shape analysis, there has been a growing awareness that many types of data are naturally invariant to certain symmetries (rotations, reparametrizations, permutations...). Such properties are directly mirrored in the intrinsic dimension of such data. These invariances cannot be faithfully transcribed by Euclidean geometry. There is therefore a growing interest in modeling such data using finer structures such as Riemannian manifolds. A second recent approach to dimension reduction consists then in generalizing existing methods to non-Euclidean data. This is known as geometric learning.In order to combine both geometric learning and manifold learning, we investigated the method called locally linear embedding, which has the specificity of being based on the notion of barycenter, a notion a priori defined in Euclidean spaces but which generalizes to Riemannian manifolds. In fact, the method called barycentric subspace analysis, which is one of those generalizing principal component analysis to Riemannian manifolds, is based on this notion as well. Here we rephrase both methods under the new notion of barycentric embeddings. Essentially, barycentric embeddings inherit the structure of most linear and non-linear dimension reduction methods, but rely on a (locally) barycentric -- affine -- model rather than a linear one.The core of our work lies in the analysis of these methods, both on a theoretical and practical level. In particular, we address the application of barycentric embeddings to two important examples in geometric learning: shapes and graphs. In addition to practical implementation issues, each of these examples raises its own theoretical questions, mostly related to the geometry of quotient spaces. In particular, we highlight that compared to standard dimension reduction methods in graph analysis, barycentric embeddings stand out for their better interpretability. In parallel with these examples, we characterize the geometry of locally barycentric embeddings, which generalize the projection computed by locally linear embedding. Finally, algorithms for geometric manifold learning, novel in their approach, complete this work
4
Lidberg, Petter. "Barycentric and harmonic coordinates." Thesis, Uppsala universitet, Algebra och geometri, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-179487.
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Hall, Stuart James. "Numerical methods and Riemannian geometry." Thesis, Imperial College London, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.538692.
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Ferreira, Ana Cristina Castro. "Riemannian geometry with skew torsion." Thesis, University of Oxford, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526550.
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Wu, Bao Qiang. "Geometry of complete Riemannian Submanifolds." Lyon 1, 1998. http://www.theses.fr/1998LYO10064.
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La géométrie rienmannienne des sous-variétés a connu ces cinquante dernières années un essor considérable, essentiellement dans le cas compact. Cette thèse a pour but de développer des outils consacrés à l'étude des sous-variétés riemanniennes complètes. Ces outils sont proches de ceux développés par Bochner et Lichnérowicz. Ils sont particulièrement adaptés aux problèmes de rigidité de certains types de sous-variétés complètes : celles qui sont à courbure moyenne constante dans un espace hyperbolique. Il est ainsi possible d'obtenir un théorème de classification de ces sous-variétés. D'autres applications sont données pour des sous-variétés totalement réelles des espaces projectifs complexes
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Boarotto, Francesco. "Topics in sub-Riemannian geometry." Doctoral thesis, SISSA, 2016. http://hdl.handle.net/20.500.11767/4881.
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This thesis is concerned with three different problems in sub-Riemannian geometry faced during my PhD. The first one is a problem in differential geometry and is about the local conformal classification of a certain class of sub-Riemannian structures. In the second one we deal with topology, and our main result establish some path-fibration properties for the Endpoint map. In the third and last problem, we begin the development of some variational calculus around critical points of the endpoint map, called abnormal controls, and we estabilish a counterpart of the classical Morse deformation techniques and of the Min-Max variational principle.
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Palmer, Ian Christian. "Riemannian geometry of compact metric spaces." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/34744.
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A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the
space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.
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Raineri, Emanuele. "Quantum Riemannian geometry of finite sets." Thesis, Queen Mary, University of London, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.414738.
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Habermann, Karen. "Geometry of sub-Riemannian diffusion processes." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/271855.
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Sub-Riemannian geometry is the natural setting for studying dynamical systems, as noise often has a lower dimension than the dynamics it enters. This makes sub-Riemannian geometry an important field of study. In this thesis, we analysis some of the aspects of sub-Riemannian diffusion processes on manifolds. We first focus on studying the small-time asymptotics of sub-Riemannian diffusion bridges. After giving an overview of recent work by Bailleul, Mesnager and Norris on small-time fluctuations for the bridge of a sub-Riemannian diffusion, we show, by providing a specific example, that, unlike in the Riemannian case, small-time fluctuations for sub-Riemannian diffusion bridges can exhibit exotic behaviours, that is, qualitatively different behaviours compared to Brownian bridges. We further extend the analysis by Bailleul, Mesnager and Norris of small-time fluctuations for sub-Riemannian diffusion bridges, which assumes the initial and final positions to lie outside the sub-Riemannian cut locus, to the diagonal and describe the asymptotics of sub-Riemannian diffusion loops. We show that, in a suitable chart and after a suitable rescaling, the small-time diffusion loop measures have a non-degenerate limit, which we identify explicitly in terms of a certain local limit operator. Our analysis also allows us to determine the loop asymptotics under the scaling used to obtain a small-time Gaussian limit for the sub-Riemannian diffusion bridge measures by Bailleul, Mesnager and Norris. In general, these asymptotics are now degenerate and need no longer be Gaussian. We close by reporting on work in progress which aims to understand the behaviour of Brownian motion conditioned to have vanishing $N$th truncated signature in the limit as $N$ tends to infinity. So far, it has led to an analytic proof of the stand-alone result that a Brownian bridge in $\mathbb{R}^d$ from $0$ to $0$ in time $1$ is more likely to stay inside a box centred at the origin than any other Brownian bridge in time $1$.
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Dunn, Corey. "Curvature homogeneous pseudo-Riemannian manifolds /." view abstract or download file of text, 2006. http://proquest.umi.com/pqdweb?did=1188874491&sid=3&Fmt=2&clientId=11238&RQT=309&VName=PQD.
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Thesis (Ph. D.)--University of Oregon, 2006.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 146-147). Also available for download via the World Wide Web; free to University of Oregon users.
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Lärz, Kordian. "Global aspects of holonomy in pseudo-Riemannian geometry." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2011. http://dx.doi.org/10.18452/16363.
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In dieser Arbeit untersuchen wir die Interaktion von Holonomie und der globalen Geometrie von Lorentzmannigfaltigkeiten und pseudo-Riemannschen Untermannigfaltigkeiten in Räumen konstanter Krümmung. Insbesondere konstruieren wir schwach irreduzible, reduzible Lorentzmetriken auf den Totalräumen von gewissen Kreisbündeln, was zu einer Konstruktionsmethode von Lorentzmannigfaltigkeiten mit vorgegebener Holonomiedarstellung führt. Danach führen wir eine Bochnertechnik für die Lorentzmannigfaltigkeiten ein, die ein nirgends verschwindendes, paralleles, lichtartiges Vektorfeld zulassen, dessen orthogonale Distribution kompakte Blätter hat. Schließlich klassifizieren wir normale Holonomiedarstellungen von raumartigen Untermannigfaltigkeiten in Räumen konstanter Krümmung und verallgemeinern die Klassifikation eine größere Klasse von Untermannigfaltigkeiten.In this thesis we study the interaction of holonomy and the global geometry of Lorentzian manifolds and pseudo-Riemannian submanifolds in spaces of constant curvature. In particular, we construct weakly irreducible, reducible Lorentzian metrics on the total spaces of certain circle bundles leading to a construction of Lorentzian manifolds with specified holonomy representations. Then we introduce a Bochner technique for Lorentzian manifolds admitting a nowhere vanishing parallel lightlike vector field whose orthogonal distribution has compact leaves. Finally, we classify normal holonomy representations of spacelike submanifolds in spaces of constant curvature and extend the classification to more general submanifolds.
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Schiappa, Ricardo. "Aspects of Riemannian geometry in quantum field theories." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/85337.
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Ghezzi, Roberta. "Almost-Riemannian Geometry from a Control Theoretical Viewpoint." Doctoral thesis, SISSA, 2010. http://hdl.handle.net/20.500.11767/4140.
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Collin, Jan-Ola. "The Existence of Riemannian Metrics on Real Vector Bundles." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-151964.
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In this thesis we present a self-contained proof of the existence of Riemannian metrics on real vector bundles.I denna uppsats presenterar vi ett självständigt bevis på existensen av Riemannskametriker på reella vektorbuntar.
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Becker, Christian. "On the Riemannian geometry of Seiberg-Witten moduli spaces." Phd thesis, [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975744771.
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Majid, Shahn, and Andreas Cap@esi ac at. "Riemannian Geometry of Quantum Groups and Finite Groups with." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi902.ps.
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Paulat, Martin [Verfasser]. "Sub-Riemannian geometry and heat kernel estimates / Martin Paulat." Kiel : Universitätsbibliothek Kiel, 2008. http://d-nb.info/1019659203/34.
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Bullo, Francesco Murray Richard M. "Nonlinear control of mechanical systems : a Riemannian geometry approach /." Diss., Pasadena, Calif. : California Institute of Technology, 1999. http://resolver.caltech.edu/CaltechETD:etd-02072008-100242.
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Nicolussi, Golo Sebastiano. "Topics in the geometry of non Riemannian lie groups." Doctoral thesis, Università degli studi di Trento, 2017. https://hdl.handle.net/11572/367715.
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This dissertation consists of an introduction and four papers. The papers deal with several problems of non-Riemannian metric spaces, such as sub-Riemannian Carnot groups and homogeneous metric spaces. The research has been carried out between the University of Trento (Italy) and the University of Jyväskylä (Finland) under the supervision of prof. F. Serra Cassano and E. Le Donne, respectively. In the following we present the abstracts of the four papers.
1) REGULARITY PROPERTIES OF SPHERES IN HOMOGENEOUS GROUPS
E. Le Donne AND S. Nicolussi Golo
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of every homogeneous distance on the Heisenberg group. In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres present cusps.
2) ASYMPTOTIC BEHAVIOR OF THE RIEMANNIAN HEISENBERG GROUP AND ITS HOROBOUNDARY
E. Le Donne, S. Nicolussi Golo, AND A. Sambusetti
The paper is devoted to the large scale geometry of the Heisenberg group H equipped with left-invariant Riemannian metrics. We prove that two such metrics have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian metric d on H there is a unique sub-Riemanniann metric d' for which d − d' goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group.
3) FROM HOMOGENEOUS METRIC SPACES TO LIE GROUPS
M. G. Cowling, V. Kivioja, E. Le Donne, S. Nicolussi Golo, AND A. Ottazzi
We study connected, locally compact metric spaces with transitive isometry groups. For all $\epsilon\in R_+$, each such space is $(1,\epsilon)$- quasi-isometric to a Lie group equipped with a left-invariant metric. Further, every metric Lie group is $(1,C)$-quasi-isometric to a solvable Lie group, and every simply connected metric Lie group is $(1,C)$-quasi-isometrically homeomorphic to a solvable-by-compact metric Lie group. While any contractible Lie group may be made isometric to a solvable group, only those that are solvable and of type (R) may be made isometric to a nilpotent Lie group, in which case the nilpotent group is the nilshadow of the group. Finally, we give a complete metric characterisation of metric Lie groups for which there exists an automorphic dilation. These coincide with the metric spaces that are locally compact, connected, homogeneous, and admit a metric dilation.
4) SOME REMARKS ON CONTACT VARIATIONS IN THE FIRST HEISENBERG GROUP
S. Nicolussi Golo
We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if $f : R^2 \rightarrow R^2$ is a $C^1$-intrinsic function, and $\nabla^f\nabla^ff = 0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.
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Nicolussi, Golo Sebastiano. "Topics in the geometry of non Riemannian lie groups." Doctoral thesis, University of Trento, 2017. http://eprints-phd.biblio.unitn.it/2668/1/SebastianoNicolussiGolo-PhDThesisUnitn.pdf.
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This dissertation consists of an introduction and four papers. The papers deal with several problems of non-Riemannian metric spaces, such as sub-Riemannian Carnot groups and homogeneous metric spaces. The research has been carried out between the University of Trento (Italy) and the University of Jyväskylä (Finland) under the supervision of prof. F. Serra Cassano and E. Le Donne, respectively. In the following we present the abstracts of the four papers.
1) REGULARITY PROPERTIES OF SPHERES IN HOMOGENEOUS GROUPS
E. Le Donne AND S. Nicolussi Golo
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an algebraic criterium in terms of the dilating automorphisms, which for example makes us conclude the regularity of every homogeneous distance on the Heisenberg group. In such a group, we analyze in more details the geometry of metric spheres. We also provide examples of homogeneous groups where spheres present cusps.
2) ASYMPTOTIC BEHAVIOR OF THE RIEMANNIAN HEISENBERG GROUP AND ITS HOROBOUNDARY
E. Le Donne, S. Nicolussi Golo, AND A. Sambusetti
The paper is devoted to the large scale geometry of the Heisenberg group H equipped with left-invariant Riemannian metrics. We prove that two such metrics have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant Riemannian metric d on H there is a unique sub-Riemanniann metric d' for which d − d' goes to zero at infinity, and we estimate the rate of convergence. As a first immediate consequence we get that the Riemannian Heisenberg group is at bounded distance from its asymptotic cone. The second consequence, which was our aim, is the explicit description of the horoboundary of the Riemannian Heisenberg group.
3) FROM HOMOGENEOUS METRIC SPACES TO LIE GROUPS
M. G. Cowling, V. Kivioja, E. Le Donne, S. Nicolussi Golo, AND A. Ottazzi
We study connected, locally compact metric spaces with transitive isometry groups. For all $\epsilon\in R_+$, each such space is $(1,\epsilon)$- quasi-isometric to a Lie group equipped with a left-invariant metric. Further, every metric Lie group is $(1,C)$-quasi-isometric to a solvable Lie group, and every simply connected metric Lie group is $(1,C)$-quasi-isometrically homeomorphic to a solvable-by-compact metric Lie group. While any contractible Lie group may be made isometric to a solvable group, only those that are solvable and of type (R) may be made isometric to a nilpotent Lie group, in which case the nilpotent group is the nilshadow of the group. Finally, we give a complete metric characterisation of metric Lie groups for which there exists an automorphic dilation. These coincide with the metric spaces that are locally compact, connected, homogeneous, and admit a metric dilation.
4) SOME REMARKS ON CONTACT VARIATIONS IN THE FIRST HEISENBERG GROUP
S. Nicolussi Golo
We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, we show that if $f : R^2 \rightarrow R^2$ is a $C^1$-intrinsic function, and $\nabla^f\nabla^ff = 0$, then the first contact variation of the sub-Riemannian area of its intrinsic graph is zero and the second contact variation is positive.
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Barilari, Davide. "Invariants, volumes and heat kernels in sub-Riemannian geometry." Doctoral thesis, SISSA, 2011. http://hdl.handle.net/20.500.11767/4631.
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Sub-Riemannian geometry can be seen as a generalization of Riemannian geometry under non-holonomic constraints. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators (see [32, 57, 70, 92] and references therein) and many problems of geometric measure theory (see for instance [18, 79]). In applications it appears in the study of many mechanical problems (robotics, cars with trailers, etc.) and recently in modern elds of research such as mathematical models of human behaviour, quantum control or motion of self-propulsed micro-organism (see for instance [15, 29, 34])
Very recently, it appeared in the eld of cognitive neuroscience to model the
functional architecture of the area V1 of the primary visual cortex, as proposed by Petitot in [87, 86], and then by Citti and Sarti in [51]. In this context, the sub-Riemannian heat equation has been used as basis to new applications in image reconstruction (see [35]).
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Gentile, Alessandro. "Geodesics and horizontal-path spaces in sub-Riemannian geometry." Doctoral thesis, SISSA, 2014. http://hdl.handle.net/20.500.11767/3901.
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Li, Chengbo. "Parametrized Curves in Lagrange Grassmannians and Sub-Riemannian Geometry." Doctoral thesis, SISSA, 2009. http://hdl.handle.net/20.500.11767/4625.
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The thesis is devoted to Differential Geometry of parametrized curves in Lagrange Grassmannians
and its applications to Optimal Control Problems and Hamiltonian Dynamics, especially
to Sub-Riemannian Geometry.
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Frost, George. "The projective parabolic geometry of Riemannian, Kähler and quaternion-Kähler metrics." Thesis, University of Bath, 2016. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.690742.
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We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.
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Wells, Matthew J. "ASPECTS OF THE GEOMETRY OF METRICAL CONNECTIONS." UKnowledge, 2009. http://uknowledge.uky.edu/gradschool_diss/749.
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Differential geometry is about space (a manifold) and a geometric structure on that space. In Riemann’s lecture (see [17]), he stated that “Thus arises the problem, to discover the matters of fact from which the measure-relations of space may be determined...”. It is key then to understand how manifolds differ from one another geometrically. The results of this dissertation concern how the geometry of a manifold changes when we alter metrical connections. We investigate how diverse geodesics are in different metrical connections. From this, we investigate a new class of metrical connections which are dependent on the class of smooth functions. Specifically, we fix a Riemannian metric and investigate the geometry of the manifold when we change the metrical connections associated with the fixed Riemannian metric. We measure the change in the Riemannian curvatures associated with this new class of metrical connections, and then give uniqueness and existence criterion for curvature of compact 2-manifolds. These results depend on the use of Hodge Theory and ultimately on the function f we choose to define a metrical connection.
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LaGatta, Tom. "Geodesics of Random Riemannian Metrics." Diss., The University of Arizona, 2010. http://hdl.handle.net/10150/193749.
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We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on R^d . We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction v, the geodesic starting from the origin in the direction v is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.
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Stavrov, Iva. "Spectral geometry of the Riemann curvature tensor /." view abstract or download file of text, 2003. http://wwwlib.umi.com/cr/uoregon/fullcit?p3095275.
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Thesis (Ph. D.)--University of Oregon, 2003.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 236-241). Also available for download via the World Wide Web; free to University of Oregon users.
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Alekseevsky, Dmitri, Andreas Kriegl, Mark Losik, Peter W. Michor, and Peter Michor@esi ac at. "The Riemannian Geometry of Orbit Spaces. The Metric, Geodesics, and." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi997.ps.
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Elsabrouty, Maha. "Riemannian geometry based blind signal separation using independent component analysis." Thesis, University of Ottawa (Canada), 2006. http://hdl.handle.net/10393/29291.
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Blind Source Separation is one of the newest and most active research areas in adaptive filtering. It represents the solution for many real situations in the audio, speech processing and telecommunication fields. The word "blind" reflects the fact that neither the source nor the mixing channel is known. This is, clearly, a more difficult situation compared to conventional adaptive filtering problems. Algorithms developed for blind separation reflect this difficulty. They possess a higher degree of sophistication compared with algorithms in other adaptive filtering approaches. The cost functions employed for blind separation are mostly based, implicitly or explicitly, on higher order statistics, while some of the adaptive algorithms developed for blind signal separation witnessed the introduction of the powerful principle of differential geometry to modify the LMS-based algorithms to what is known as the natural-gradient update.
The aim of this work is to generalize differential geometry algorithms for different cost functions of blind signal separation and provide new faster converging RLS-based and Newton-based algorithms using the natural gradient update. The thesis starts by providing a thorough review of the existing solutions developed in the literature for blind separation. This is followed by a study of the mathematical basis of Riemannian geometry and providing an engineering insight into the intrinsic geometry of curved spaces and its relation to optimization in adaptive filtering. Several new update algorithms are then developed throughout the thesis. They are structured to perform at faster convergence rates even in difficult mixing situations. These algorithms provide significantly improved performance in comparison with existing algorithms and are suitable for on-line applications.
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Malchiodi, Andrea. "Existence and multiplicity results for some problems in Riemannian geometry." Doctoral thesis, SISSA, 2000. http://hdl.handle.net/20.500.11767/4627.
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Onodera, Mitsuko. "Study of rigidity problems for C2[pi]-manifolds." Sendai : Tohoku Univ, 2006. http://www.gbv.de/dms/goettingen/52860726X.pdf.
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Rizzi, Luca. "The curvature of optimal control problems with applications to sub-Riemannian geometry." Doctoral thesis, SISSA, 2014. http://hdl.handle.net/20.500.11767/4841.
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Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism).
In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated.
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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.Committee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ian
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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.Ph. D.
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Persson, Nicklas. "Shortest paths and geodesics in metric spaces." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-66732.
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This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.
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Feppon, Florian (Florian Jeremy). "Riemannian geometry of matrix manifolds for Lagrangian uncertainty quantification of stochastic fluid flows." Thesis, Massachusetts Institute of Technology, 2017. http://hdl.handle.net/1721.1/111041.
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Thesis: S.M., Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2017.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 119-129).
This work focuses on developing theory and methodologies for the analysis of material transport in stochastic fluid flows. In a first part, two dominant classes of techniques for extracting Lagrangian Coherent Structures are reviewed and compared and some improvements are suggested for their pragmatic applications on realistic high-dimensional deterministic ocean velocity fields. In the stochastic case, estimating the uncertain Lagrangian motion can require to evaluate an ensemble of realizations of the flow map associated with a random velocity flow field, or equivalently realizations of the solution of a related transport partial differential equation. The Dynamically Orthogonal (DO) approximation is applied as an efficient model order reduction technique to solve this stochastic advection equation. With the goal of developing new rigorous reduced-order advection schemes, the second part of this work investigates the mathematical foundations of the method. Riemannian geometry providing an appropriate setting, a framework free of tensor notations is used to analyze the embedded geometry of three popular matrix manifolds, namely the fixed rank manifold, the Stiefel manifold and the isospectral manifold. Their extrinsic curvatures are characterized and computed through the study of the Weingarten map. As a spectacular by-product, explicit formulas are found for the differential of the truncated Singular Value Decomposition, of the Polar Decomposition, and of the eigenspaces of a time dependent symmetric matrix. Convergent gradient flows that achieve related algebraic operations are provided. A generalization of this framework to the non-Euclidean case is provided, allowing to derive analogous formulas and dynamical systems for tracking the eigenspaces of non-symmetric matrices. In the geometric setting, the DO approximation is a particular case of projected dynamical systems, that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. It is obtained that the error committed by the DO approximation is controlled under the minimal geometric condition that the original solution stays close to the low-rank manifold. The last part of the work focuses on the practical implementation of the DO methodology for the stochastic advection equation. Fully linear, explicit central schemes are selected to ensure stability, accuracy and efficiency of the method. Riemannian matrix optimization is applied for the dynamic evaluation of the dominant SVD of a given matrix and is integrated to the DO time-stepping. Finally the technique is illustrated numerically on the uncertainty quantification of the Lagrangian motion of two bi-dimensional benchmark flows.
by Florian Feppon.
S.M.
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Bizi, Nadir. "Semi-riemannian noncommutative geometry, gauge theory, and the standard model of particle physics." Thesis, Sorbonne université, 2018. http://www.theses.fr/2018SORUS413/document.
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Dans cette thèse, nous nous intéressons à la géométrie non-commutative - aux triplets spectraux en particulier - comme moyen d'unifier gravitation et modèle standard de la physique des particules. Des triplets spectraux permettant une telle unification on déjà été construits dans le cas des variétés riemanniennes. Il s'agit donc ici de généraliser au cas des variétés semi-riemanniennes, et d'appliquer ensuite au cas lorentzien, qui est d'une importance particulière en physique. C'est ce que nous faisons dans la première partie de la thèse, ou le passage du cas riemannien au cas semi-riemannien nous oblige à nous intéresser à des espaces vectoriels de signatures indéfinies (et non définies positives), dits espaces de Krein. Ceci est une conséquence de notre étude des algèbres de Clifford indéfinies et des structures Spin sur variétés semi-riemanniennes. Nous généralisons ensuite les triplets spectraux en triplets dits indéfinis en conséquence de cela. Dans la deuxième partie de la thèse, nous appliquons le formalisme des formes différentielles non-commutatives à nos triplets indéfinis pour formuler des théories de jauge non-commutatives sur espace-temps lorentzien. Nous montrons ensuite comment obtenir le modèle standardThe subject of this thesis is noncommutative geometry - more specifically spectral triples - and how it can be used to unify General Relativity with the Standard Model of particle physics. This unification has already been achieved with spectral triples for Riemannian manifolds. The main concern of this thesis is to generalize this construction to semi-Riemannian manifolds generally, and Lorentzian manifolds in particular. The first half of this thesis will thus be dedicated to the transition from Riemannian to semi-Riemannian manifolds. This entails a study of Clifford algebras for indefinite vector spaces and Spin structures on semi-Riemannian manifolds. An important consequence of this is the introduction of complex vector spaces of indefinite signature. These are the so-called Krein spaces, which will enable us to generalize spectral triples to indefinite spectral triples. In the second half of this thesis, we will apply the formalism of noncommutative differential forms to indefinite spectral triples to construct noncommutative gauge theories on Lorentzian spacetimes. We will then demonstrate how to recover the Standard Model
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Baspinar, Emre <1988>. "Minimal Surfaces in Sub-Riemannian Structures and Functional Geometry of the Visual Cortex." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amsdottorato.unibo.it/8661/7/thesis_baspinar_for_submission.pdf.
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We develop geometrical models of vision consistent with the characteristics of the visual cortex and study geometric flows in the relevant model geometries. We provide a novel sub-Riemannian model of the primary visual cortex, which models orientation-frequency selective phase shifted cortex cell behavior and the associated horizontal connectivity. We develop an image enhancement algorithm using sub-Riemannian diffusion
and Laplace-Beltrami flow in the model framework. We provide two geometric models for multi-scale orientation map and orientation-frequency preference map construction which employ Bargmann transform in high dimensional cortical spaces. We prove the uniqueness of the solution to sub-Riemannian mean curvature flow equation in the Heisenberg group geometry. An iterative diffusion process followed by a maximum selection mechanism was proposed by Citti and Sarti in the sub-Riemannian setting of the roto-translation group. They conjectured that this two-fold procedure is equivalent to a mean curvature flow. However a complete proof was missing, even in the Euclidean setting. We prove in the Euclidean
setting that this two fold procedure is equivalent to mean curvature flow.
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Rupassara, Rupassarage Upul Hemakumara. "Joint exit time and place distribution for Brownian motion on Riemannian manifolds." OpenSIUC, 2019. https://opensiuc.lib.siu.edu/dissertations/1720.
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This dissertation discusses the time and place that Brownian motion on a Riemannian manifold first exit a normal ball of small radius. A general procedure is given for computing asymptotic expansions of joint moments of the first exit time and place random variables as the radius of the geodesic ball decreases to zero. The asymptotic expansion of the joint Laplace transform of exit time and spherical harmonics of exit position is derived for a ball of small radius. A generalized Pizetti’s formula is used to expand the solution of the related partial differential equations. These expansions are represented in terms of curvature in the manifold. Asymptotic Independence Conditions (AIC) and Asymptotic Uncorrelated Conditions (AUC) are defined for the joint distributions of exit time and place. Computations using the methods developed in this work demonstrate that AIC and AUC produce the same curvature conditions up to a certain level of asymptotics. It is conjectured that AUC implies AIC. Further, a generalized method is given for computing the Laplace transform, and therefore the moments of the exit time. This work is related to and also extends the work of M. Liao and H. R. Hughes in stochastic geometric analysis.
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McMahon, Joseph Brian. "Geometry and Mechanics of Growing, Nonlinearly Elastic Plates and Membranes." Diss., The University of Arizona, 2009. http://hdl.handle.net/10150/194028.
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Until the twentieth century, theories of elastic rods and shells arose from collections of geometric and mechanical assumptions and approximations. These theories often lacked internal consistency and were appropriate for highly proscribed and sometimes unknown geometries and deformation sizes. The pioneering work of Truesdell, Antman, and others converted mechanical intuition into rigorous mathematical statements about the physics and mechanics of rods and shells. The result is the modern, geometrically exact theory of finite deformations of rods and shells.In the latter half of the twentieth century, biomechanics became a major focus of both experimental and theoretical mechanics. The genesis of residual stress by non-elastic growth has significant impact on the shape and mechanical properties of soft tissues. Inspired by the geometry of blood vessels and adopting a formalism found in elasto-plasticity, mechanicians have produced rigorous and applied results on the effect of growth on finite elastic deformations of columns and hollow tubes. Less attention has been paid to shells.A theory of growing elastic plates has been constructed in the context of linear elasticity. It harnessed many results in the theory of Riemann surfaces and has produced solutions that are surprisingly similar to experimental observations. Our intention is to provide a finite-deformation alternative by combining growth with the geometrically exact theory of shells. Such a theory has a clearer and more rigorous foundation, and it is applicable to thicker structures than is the case in the current theory of growing plates.This work presents the basic mathematical tools required to construct this alternative theory of finite elasticity of a shell in the presence of growth. We make clear that classical elasticity can be viewed in terms of three-dimensional Riemannian geometry, and that finite elasticity in the presence of growth must be considered in this way. We present several examples that demonstrate the viability and tractability of this approach.
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Leijon, Rasmus. "The Einstein Field Equations : on semi-Riemannian manifolds, and the Schwarzschild solution." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-61321.
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Semi-Riemannian manifolds is a subject popular in physics, with applications particularly to modern gravitational theory and electrodynamics. Semi-Riemannian geometry is a branch of differential geometry, similar to Riemannian geometry. In fact, Riemannian geometry is a special case of semi-Riemannian geometry where the scalar product of nonzero vectors is only allowed to be positive. This essay approaches the subject from a mathematical perspective, proving some of the main theorems of semi-Riemannian geometry such as the existence and uniqueness of the covariant derivative of Levi-Civita connection, and some properties of the curvature tensor. Finally, this essay aims to deal with the physical applications of semi-Riemannian geometry. In it, two key theorems are proven - the equivalenceof the Einstein field equations, the foundation of modern gravitational physics, and the Schwarzschild solution to the Einstein field equations. Examples of applications of these theorems are presented.
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Cheung, Leung-Fu. "Geometric properties of stable noncompact constant mean curvature surfaces." Bonn : [s.n.], 1991. http://catalog.hathitrust.org/api/volumes/oclc/26531351.html.
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Hainz, Stefan. "Eine Riemannsche Betrachtung des Reeb-Flusses." Bonn : Mathematisches Institut der Universität, 2006. http://catalog.hathitrust.org/api/volumes/oclc/173261156.html.
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Reach, Andrew McCaleb. "Smooth Interactive Visualization." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/78848.
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Information visualization is a powerful tool for understanding large
datasets. However, many commonly-used techniques in information
visualization are not C^1 smooth, i.e. when represented as a
function, they are either discontinuous or have a discontinuous first
derivative. For example, histograms are a
non-smooth visualization of density. Not only are histograms
non-smooth visually, but they are also non-smooth over their parameter
space, as they change abruptly in response to smooth change of bin
width or bin offset. For large data visualization, histograms are
commonly used in place of smooth alternatives, such as kernel density
plots, because histograms can be constructed from data cubes, allowing histograms to be constructed quickly for large datasets.
Another example of a non-smooth technique in information visualization
is the commonly-used transition approach to animation. Although
transitions are designed to create smooth animations, the transition
technique produces animations that have velocity discontinuities if
the target is changed before the transition has finished. The smooth
and efficient zooming and panning technique also shares this
problem---the animations produced are smooth while in-flight, but they
have velocity discontinuities at the beginning and end and of the
animation as well as velocity discontinuities when interrupted.
This dissertation applies ideas from signal processing to construct
smooth alternatives to these non-smooth techniques. To visualize
density for large datasets, we propose BLOCs, a smooth alternative to
data cubes that allows kernel density plots to be constructed quickly
for large datasets after an initial preprocessing step. To create
animations that are smooth even when interrupted, we present LTI
animation, a technique that uses LTI filters to create animations that
are smooth, even when interrupted. To create zooming and panning
animations that are smooth, even when interrupted, we generalize
signal processing systems to Riemannian manifolds, resulting in
smooth, efficient, and interruptible animations.Ph. D.
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ROITBERG, ALICE. "Gross-Pitaevskii hydrodynamics in Riemannian manifolds and application in Black Hole cosmology." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2023. https://hdl.handle.net/10281/404710.
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In questa tesi ci si propone di analizzare le implicazioni fisiche della geometria dello spazio ambiente nel contesto dei condensati di Bose-Einstein (BEC) e le possibili applicazioni nell’ambito dei modelli analogici della cosmologia dei buchi neri. A tal fine si deriva la formulazione idrodinamica dell'equazione di Gross-Pitaevskii (GPE) nel caso di una generica varietà Riemanniana e si osserva la comparsa di una nuova forza, che dipende essenzialmente da due parametri: la geometria della varietà e le derivate prime del profilo di densità. Si studiano le condizioni stazionarie in relazione alla presenza di varietà a curvatura scalare negativa. Analizzando tali varietà si stabilisce una relazione esplicita tra le superfici a curvatura negativa e l'equazione di seno-Gordon, che risulta un'approssimazione della GPE nel caso di accoppiamento di fasi.
Assumendo condizioni stazionarie, si ottiene un nuovo tipo di equazioni di Einstein e si è spinti a ricercare altri legami tra le equazioni che governano i condensati e la cosmologia. A tal fine si considerano i BEC relativistici, che vengono utilizzati nello studio del comportamento dell'universo primordiale e della sua espansione. Facendo uso delle conoscenze ottenute nel caso di varietà Riemanniane generiche, otteniamo nuove equazioni di Einstein nel caso multi-dimensionale. Successivamente, si considerano i modelli analogici utilizzati per lo studio della formazione di buchi neri e per il calcolo della radiazione di Hawking. Attraverso un processo di linearizzazione si nota come sia possibile far emergere una metrica acustica Lorentziana che governi il moto delle fluttuazioni della fase; a questo scopo si considera il caso di un vortice dritto che presenta un profilo di densità in cui le derivate prime assumono un valore massimo all’interno del tubo vorticoso e la geometria dello spazio ambiente diventa rilevante. In questa situazione si scopre che è effettivamente possibile far emergere una metrica Lorentziana, e si propongono alcune approssimazioni utili per la sua determinazione esplicita.
Infine, vengono presentate alcune osservazioni conclusive su possibili direzioni di ricerca future, quali lo studio dell'evoluzione delle superfici isofase in casi relativistici e lo studio dei condensati sottoposti a torsione.In this thesis we analyze the physical implications of the geometry of the ambient space in the context of Bose-Einstein condensates (BECs) and possible applications to the field of analogue models in the cosmology of black holes. To this end we derive the hydrodynamic formulation of the Gross-Pitaevskii equation (GPE) in the case of a generic Riemannian manifold. We observe the appearance of a new force, which essentially depends on two parameters: the geometry of the manifold and the first derivatives of the density profile. The stationary conditions are studied in relation to the presence of manifolds with negative scalar curvature. By analyzing these manifolds, an explicit relationship is established between the negatively curved surfaces and the sine-Gordon equation, which results in an approximation of the GPE in the presence of phase coupling. By assuming stationary conditions, we obtain a new type of Einstein field equations and we look for other possible connections between the equations governing condensates and cosmology. For this purpose, we consider relativistic BECs, that are used in the study of the early universe and its expansion, and we obtain Einstein equation in the multi-dimensional case. Then we consider the analogue models used for the study of the formation of black holes and for the calculation of Hawking radiation. Through a linearization process it is possible to derive a Lorentzian acoustic metric for the phase fluctuations; for this purpose, we consider the case of a straight vortex defect with a density profile where the first derivatives have maximum value inside the vortex tube and the geometry of the ambient space becomes relevant. In this situation it turns out that it is possible to determine a Lorentzian metric, and some useful approximations are proposed for its explicit computation. Finally, some concluding remarks are presented on possible future research directions, given by the study of the evolution of isophase surfaces in relativistic cases, and the study of condensates subject to twist.
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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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Ruscelli, Francesco. "On the Palatini formulation of general relativity." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23693/.
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In questo lavoro presentiamo la formulazione di Palatini della Relatività Generale. Per fare ciò, discutiamo le nozioni di base della teoria delle varietà differenziabili e della geometria (pseudo-)Riemanniana. In particolare, ci concentriamo sui fibrati vettoriali e sul concetto di connessione. Quest'ultimo è analizzato sia nel caso particolare di una connessione affine, ovvero sul fibrato tangente ad una varietà, sia nel caso generale di una connessione su un fibrato vettoriale arbitrario.
Mentre la geometria Riemanniana e la Relatività Generale sono state storicamente sviluppate attraverso il calcolo tensoriale, noi seguiamo l'approccio di cui Elie Cartan, nel XX secolo, fu pioniere. Questo approccio individua le forme differenziali come oggetti fondamentali della teoria e risulta fondamentale nella costruzione dell'azione di Palatini. Di conseguenza, dedichiamo ampio spazio alla descrizione della connessione come forma differenziale, insieme alle forme di curvatura e di torsione.
Prima di introdurre il formalismo di Palatini, richiamiamo brevemente le idee fisiche che hanno ispirato la Relatività Generale, come il Principio di Equivalenza, e mostriamo come si possano estrapolare le equazioni di Einstein a partire dall'equazione di Poisson per il potenziale gravitazionale. Infine, utilizziamo il formalismo di Cartan per derivare l'azione di Palatini e mostriamo come dalla variazione di quest'ultima sia possibile ritrovare le equazioni di Einstein.
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Röttgen, Nena-Maria [Verfasser], and Victor [Akademischer Betreuer] Bangert. "Existence of periodic orbits in Riemannian and contact geometry = Existenz periodischer Orbits in Riemannscher und Kontaktgeometrie." Freiburg : Universität, 2014. http://d-nb.info/1123482098/34.
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