Dissertations / Theses on the topic 'Géométrie riemannienne et barycentrique'
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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.
Full textAn 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
Niang, Athoumane. "Sur quelques problèmes en géométrie équiaffine et en géométrie semi-riemannienne." Montpellier 2, 2005. http://www.theses.fr/2005MON20043.
Full textCharuel, Xavier. "Courbes et hypersurfaces nulles en géométrie pseudo-Riemannienne." Nancy 1, 2003. http://docnum.univ-lorraine.fr/public/SCD_T_2003_0005_CHARUEL.pdf.
Full textIn this thesis, we study "degenerate" (or "null") submanifolds of pseudo-riemannian manifolds, for which the restriction of the pseudo-riemannian structure of the ambiant manifold degenerate on the submanifold. In the first part, we build a generalized Frénet 's frame in pseudo-riemannian manifolds. In the second part, we generalize our construction to other situations, such as symplectic manifolds, or pseudo-kaehlerian manifolds. Finally, in the last part of this thesis, we study totally geodesic degenerate hypersurfaces in pseudo-riemannian manifolds. We find invariants relative to the induced structure on the hypersurface, and use them to build local coordinate systems adapted to the geometry of the hypersurface
Charlot, Grégoire. "Géométrie sous-riemannienne de contact et de quasi-contact." Dijon, 2001. http://www.theses.fr/2001DIJOS030.
Full textHumbert, Emmanuel. "Inégalités optimales de types Nash et Sobolev en géométrie riemannienne." Paris 6, 2000. http://www.theses.fr/2000PA066218.
Full textRoth, Julien. "Rigidité des hypersurfaces en géométrie riemannienne et spinorielle : Aspect extrinsèque et intrinsèque." Phd thesis, Université Henri Poincaré - Nancy I, 2006. http://tel.archives-ouvertes.fr/tel-00120756.
Full textJanin, Gabriel. "Contrôle optimal et applications au transfert d'orbite et à la géométrie presque-riemannienne." Phd thesis, Université de Bourgogne, 2010. http://tel.archives-ouvertes.fr/tel-00633197.
Full textFrancoeur, Dominik. "Géométrie de Cartan et pré-géodésiques de type lumière." Mémoire, Université de Sherbrooke, 2014. http://savoirs.usherbrooke.ca/handle/11143/5297.
Full textArguillere, Sylvain. "Géométrie sous-riemannienne en dimension infinie et applications à l'analyse mathématique des formes." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066144/document.
Full textThis manuscript is dedicated to the study of infinite dimensional sub-Riemannian geometry and its applications to shape analysis using dieomorphic deformations. The first part is a detailed summary of our work, while the second part combines the articles we wrote during the last three years. We first extend the framework of sub- Riemannian geometry to infinite dimensions, establishing conditions that ensure the existence of a Hamiltonian geodesic flow. We then apply these results to strong right- invariant sub-Riemannian structures on the group of diffeomorphisms of a manifold. We then define rigorously the abstract concept shape spaces. A shape space is a Banach manifold on which the group of diffeomorphisms of a manifold acts in a way that satisfy certain properties. We then define several sub-Riemannian structures on these shape spaces using this action, and study these. Finally, we add constraints to the possible deformations, and formulate shape analysis problems in an infinite dimensional control theoritic framework. We prove a Pontryagin maximum principle adapted to this context, establishing the constrained geodesic equations. Algorithms for fin- ding optimal deformations are then developped, supported by numerical simulations. These algorithms extend and unify previously established methods in shape analysis
Choné, Philippe. "Étude de quelques problèmes variationnels intervenant en géométrie riemannienne et en économie mathématique." Toulouse 1, 1999. http://www.theses.fr/1999TOU10020.
Full textIn the first part of this thesis, we consider a critical point u of a conformally invariant functional on a two-dimensional domain. We show that if u is a priori assumed to be bounded, then u is smooth up to the boundary of the domain. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three dimensional compact Riemannian manifold. The variational problems studied in the second part are motivated by economic issues, namely non-linear pricing by a monopolist or a duopolist. The problem consists in maximizing a functional over the cone of convex functions. We give a sufficient condition for the convexity constraint to be active. This condition does hold in many common situations in economics. Typically, in a two-dimensional problem, there exists an area where the rank of the hessian of the solution is 1. We write the Euler equation of the problem and derive the + sweeping conditions. We explain how to use these conditions to compute the solution. This method, however, requires some prior knowledge of the solution. We therefore study the numerical approximation of the problem. We show how to apply some simple finite-elements methods to the problem. There is, however, a strong theoretical obstruction to the convergence of these methods (in dimension greater than 2). Finally we consider duopoly models that involve non-concave and non-coercive functionals. We study best reply maps and Nash equilibria in these models
Gris, Barbara. "Approche modulaire sur les espaces de formes, géométrie sous-riemannienne et anatomie computationnelle." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLN069/document.
Full textThis thesis is dedicated to the development of a new deformation model to study shapes. Deformations, and diffeormophisms in particular, have played a tremendous role in the field of statistical shape analysis, as a proxy to measure and interpret differences between similar objects but with different shapes. Diffeomorphisms usually result from the integration of a flow of regular velocity fields, whose parameters have not enabled so far a full control of the local behaviour of the deformation. We propose a new model in which velocity fields are built on the combination of a few local and interpretable vector fields. These vector fields are generated thanks to a structure which we name deformation module. Deformation modules generate vector fields of a particular type (e.g. a scaling) chosen in advance: they allow to incorporate a constraint in the deformation model. These constraints can correspond either to an additional knowledge one would have on the shapes under study, or to a point of view from which one would want to study these shapes. In a first chapter we introduce this notion of deformation module and we give several examples to show how diverse they can be. We also explain how one can easily build complex deformation modules adapted to complex constraints by combining simple deformation modules. Then we introduce the construction of modular large deformations as flow of vector fields generated by a deformation module. Vector fields generated by a deformation module are parametrized by two variables: a geometrical one named geometrical descriptor and a control one. We build large deformations so that the geometrical descriptor follows the deformation of the ambient space. Then defining a modular large deformation corresponds to defining an initial geometrical descriptor and a trajectory of controls. We also associate a notion of cost for each couple of geometrical descriptor and control. In a second chapter we explain how we can use a given deformation module to study data shapes. We first build a sub-Riemannian structure on the space defined as the product of the data shape space and the space of geometrical descriptors. The sub-Riemannian metric comes from the chosen cost: we equip the new (shape) space with a chosen metric, which is not in general the pull-back of a metric on vector fields but takes into account the way vector fields are built with the chosen constraints. Thanks to this structure we define a sub-Riemannian distance on this new space and we show the existence, under some mild assumptions, of geodesics (trajectories whose length equals the distance between the starting and ending points). The study of geodesics amounts to an optimal control problem, and they can be estimated thanks to an Hamiltonian framework: in particular we show that they can be parametrized by an initial variable named momentum. Afterwards we introduce optimal modular large deformations transporting a source shape into a target shape. We also define the modular atlas of a population of shapes which is made of a mean shape, and one modular large deformation per shape. In the discussion we study an alternative model where geodesics are parametrized in lower dimension. In a third chapter we present the algorithm that was implemented in order to compute these modular large deformations and the gradient descent to estimate the optimal ones as well as mean shapes. In a last chapter we introduce several numerical examples thanks to which we study specific aspects of our model. In particular we show that the choice of the used deformation module influences the form of the estimated mean shape, and that by choosing an adapted deformation module we are able to perform in a satisfying and robust way simultaneously rigid and non linear registration. In the last example we study shapes without any prior knowledge, then we use a module corresponding to weak constraints and we show that the atlas computation still gives interesting results
Andreadis, Ioannis, and Fernand Pelletier. "Contribution à l'étude des singularités en géométrie symplectique et pseudo-riemannienne en dimension infinie." Chambéry, 1995. http://www.theses.fr/1995CHAMS002.
Full textLouis, Maxime. "Méthodes numériques et statistiques pour l'analyse de trajectoire dans un cadre de géométrie Riemannienne." Electronic Thesis or Diss., Sorbonne université, 2019. http://www.theses.fr/2019SORUS570.
Full textThis PhD proposes new Riemannian geometry tools for the analysis of longitudinal observations of neuro-degenerative subjects. First, we propose a numerical scheme to compute the parallel transport along geodesics. This scheme is efficient as long as the co-metric can be computed efficiently. Then, we tackle the issue of Riemannian manifold learning. We provide some minimal theoretical sanity checks to illustrate that the procedure of Riemannian metric estimation can be relevant. Then, we propose to learn a Riemannian manifold so as to model subject's progressions as geodesics on this manifold. This allows fast inference, extrapolation and classification of the subjects
Pinoy, Alan. "Géométrie asymptotiquement hyperbolique complexe et contraintes de courbure." Thesis, Université de Montpellier (2022-….), 2022. http://www.theses.fr/2022UMONS024.
Full textIn this thesis, we investigate the asymptotic geometric properties a class of complete and non compact Kähler manifolds we call asymptotically locally complex hyperbolic manifolds.The local geometry at infinity of such a manifold is modeled on that of the complex hyperbolic space, in the sense that its curvature is asymptotic to that of the model space.Under natural geometric assumptions, we show that this constraint on the curvature ensures the existence of a rich geometry at infinity: we can endow it with a strictly pseudoconvex CR boundary at infinity
Bouchard, Florent. "Géométrie et optimisation riemannienne pour la diagonalisation conjointe : application à la séparation de sources d'électroencéphalogrammes." Thesis, Université Grenoble Alpes (ComUE), 2018. http://www.theses.fr/2018GREAS030/document.
Full textThe approximate joint diagonalisation of a set of matrices allows the solution of the blind source separation problem and finds several applications, for instance in electroencephalography, a technique for measuring brain activity.The approximate joint diagonalisation is formulated as an optimization problem with three components: the choice of the criterion to be minimized, the non-degeneracy constraint on the solution and the solving algorithm.Existing approaches mainly consider two criteria, the least-squares and the log-likelihood.They are specific to a constraint and are limited to only one type of solving algorithms.In this thesis, we propose to formulate the approximate joint diagonalisation problem in a geometrical fashion, which generalizes previous works and allows the definition of new criteria, particularly those linked to information theory.We also propose to exploit Riemannian optimisation and we define tools that allow to have the three components varying independently, creating in this way new methods and revealing the influence of the choice of the model.Numerical experiments on simulated data as well as on electroencephalographic recordings show that our approach by means of Riemannian optimisation gives results that are competitive as compared to existing methods.They also indicate that the two traditional criteria do not perform best in all situations
Gaye, Moussa. "Quelques problèmes d'analyse géométrique en géométrie quasi-Riemannienne et d'analyse de stabilité des systèmes à commutation linéaires." Palaiseau, Ecole polytechnique, 2014. http://www.theses.fr/2014EPXX0046.
Full textKalunga, Emmanuel. "Vers des interfaces cérébrales adaptées aux utilisateurs : interaction robuste et apprentissage statistique basé sur la géométrie riemannienne." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLV041/document.
Full textIn the last two decades, interest in Brain-Computer Interfaces (BCI) has tremendously grown, with a number of research laboratories working on the topic. Since the Brain-Computer Interface Project of Vidal in 1973, where BCI was introduced for rehabilitative and assistive purposes, the use of BCI has been extended to more applications such as neurofeedback and entertainment. The credit of this progress should be granted to an improved understanding of electroencephalography (EEG), an improvement in its measurement techniques, and increased computational power.Despite the opportunities and potential of Brain-Computer Interface, the technology has yet to reach maturity and be used out of laboratories. There are several challenges that need to be addresses before BCI systems can be used to their full potential. This work examines in depth some of these challenges, namely the specificity of BCI systems to users physical abilities, the robustness of EEG representation and machine learning, and the adequacy of training data. The aim is to provide a BCI system that can adapt to individual users in terms of their physical abilities/disabilities, and variability in recorded brain signals.To this end, two main avenues are explored: the first, which can be regarded as a high-level adjustment, is a change in BCI paradigms. It is about creating new paradigms that increase their performance, ease the discomfort of using BCI systems, and adapt to the user’s needs. The second avenue, regarded as a low-level solution, is the refinement of signal processing and machine learning techniques to enhance the EEG signal quality, pattern recognition and classification.On the one hand, a new methodology in the context of assistive robotics is defined: it is a hybrid approach where a physical interface is complemented by a Brain-Computer Interface (BCI) for human machine interaction. This hybrid system makes use of users residual motor abilities and offers BCI as an optional choice: the user can choose when to rely on BCI and could alternate between the muscular- and brain-mediated interface at the appropriate time.On the other hand, for the refinement of signal processing and machine learning techniques, this work uses a Riemannian framework. A major limitation in this filed is the EEG poor spatial resolution. This limitation is due to the volume conductance effect, as the skull bones act as a non-linear low pass filter, mixing the brain source signals and thus reducing the signal-to-noise ratio. Consequently, spatial filtering methods have been developed or adapted. Most of them (i.e. Common Spatial Pattern, xDAWN, and Canonical Correlation Analysis) are based on covariance matrix estimations. The covariance matrices are key in the representation of information contained in the EEG signal and constitute an important feature in their classification. In most of the existing machine learning algorithms, covariance matrices are treated as elements of the Euclidean space. However, being Symmetric and Positive-Definite (SPD), covariance matrices lie on a curved space that is identified as a Riemannian manifold. Using covariance matrices as features for classification of EEG signals and handling them with the tools provided by Riemannian geometry provide a robust framework for EEG representation and learning
Elmir, Chady. "Constante systolique et variétés plates." Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2009. http://tel.archives-ouvertes.fr/tel-00439914.
Full textGicquaud, Romain. "Etude de quelques problèmes d'analyse et de géométrie sur les variétés asymptotiquement hyperboliques." Montpellier 2, 2009. http://www.theses.fr/2009MON20101.
Full textThis thesis is divided in two parts. In the first part, we study the compactification of asymptotically locally hyperbolic manifolds, that is to say non-compact Riemannian manifolds whose sectional curvature tends to -1 at infinity. We show how the asymptotic behavior of the curvature and of its covariant derivatives influences the regularity of the compactified metric. In the Einstein case, we prove that the estimate on the sectional curvature implies the control of all covariant derivatives of the Riemann tensor, we give a conjecture on the behavior at infinity of the sectional curvature and give some demo tracks. The second part deals with the constraint equations in general relativity on an asymptotically hyperbolic manifold. First, we give a construction of solutions to these equations containing apparent horizons using the conformal method. Then we study the problem of their linearization-stability. We show in particular that initial data corresponding to empty space-times are linearization-stable in a certain range of weight. For larger weights, we show that these equations become unstable
Schäfer, Lars. "Geometrie tt* et applications pluriharmoniques." Nancy 1, 2006. http://www.theses.fr/2006NAN10041.
Full textIn this work we introduce the real differential geometric notion of a tt*-bundle (E,D,S), a metric tt*-bundle (E,D,S,g) and a symplectic tt*-bundle (E,D,S,omega) on an abstract vector bundle E over an almost complex manifold (M,J). With this notion we construct, generalizing Dubrovin, a correspondence between metric tt*-bundles over complex manifolds (M,J) and admissible pluriharmonic maps from (M,J) into the pseudo-Riemannian symmetric space GL(r,R)/O(p,q) where (p,q) is the signature of the metric g. Moreover, we show a rigidity result for tt*-bundles over compact Kähler manifolds and we obtain as application a special case of Lu's theorem. In addition we study solutions of tt*-bundles (TM,D,S) on the tangent bundle TM of (M,J) and characterize an interesting class of these solutions which contains special complex manifolds and flat nearly Kähler manifolds. We analyze which elements of this class admit metric or symplectic tt*-bundles. Further we consider solutions coming from varitations of Hodge structures (VHS) and harmonic bundles. Applying our correspondence to harmonic bundles we generalize a correspondence given by Simpson. Analyzing the associated pluriharmonic maps we obtain roughly speaking for special Kähler manifolds the dual Gauss map and for VHS of odd weight the period map. In the case of non-integrable complex structures, we need to generalize the notions of pluriharmonic maps and some results. Apart from the rigidity result we generalize all above results to para-complex geometry
Trélat, Emmanuel. "Etude asymptotique et transcendance de la fonctionvaleur en contrôle optimal. Catégorie log-exp en géométrie sous-Riemannienne dans le cas Martinet." Phd thesis, Université de Bourgogne, 2000. http://tel.archives-ouvertes.fr/tel-00086511.
Full texttrajectoires anormales en théorie du contrôle optimal.
Après avoir rappelé quelques résultats fondamentaux en contrôle
optimal, on étudie l'optimalité des
anormales pour des systèmes affines mono-entrée avec contrainte
sur le contrôle, d'abord pour le problème du temps optimal, puis
pour un coût quelconque à temps final fixé ou non.
On étend cette théorie aux
systèmes sous-Riemanniens de rang 2, montrant qu'on se ramène
à un système affine du type précédent.
Ces résultats montrent que,
sous des conditions générales, une trajectoire anormale est
\it{isolée} parmi toutes les solutions du système ayant les mêmes
conditions aux limites, et donc \it{localement optimale}, jusqu'à
un premier point dit \it{conjugué} que l'on peut caractériser.
On s'intéresse ensuite
au comportement asymptotique et à la
régularité de la fonction valeur associée à un système affine
analytique avec un coût quadratique. On montre que, en
l'absence de trajectoire
anormale minimisante, la fonction valeur est
\it{sous-analytique et continue}. S'il existe une anormale
minimisante, on sort de la catégorie sous-analytique en général,
notamment en géométrie sous-Riemannienne. La présence d'une
anormale minimisante est responsable de la \it{non-propreté} de
l'application exponentielle, ce qui provoque un phénomène de
\it{tangence} des ensembles de niveaux de la fonction valeur par
rapport à la direction anormale. Dans le cas affine mono-entrée
ou sous-Riemannien de rang 2, on décrit précisément ce
contact, et on en déduit une partition de la
sphère sous-Riemannienne au voisinage de l'anormale
en deux secteurs appelés \it{secteur
$L^\infty$} et \it{secteur $L^2$}.\\
La question de transcendance est étudiée dans le cas
sous-Riemannien de Martinet où la distribution est
$\Delta=\rm{Ker }(dz-\f{y^2}{2}dx)$. On montre que
pour une métrique générale graduée d'ordre $0$~:
$g=(1+\alpha y)^2dx^2+(1+\beta x+\gamma y)^2dy^2$,
les sphères de petit rayon
\it{ne sont pas sous-analytiques}. Dans le cas général
intégrable où $g=a(y)dx^2+c(y)dy^2$, avec $a$ et $c$ analytiques,
les sphères de Martinet appartiennent à la
\it{catégorie log-exp}.
Lawn-Paillusseau, Marie-Amelie. "Méthodes Spinorielles et géométrie para-complexe et para-quaternionique en théorie des sous-variétés." Phd thesis, Université Henri Poincaré - Nancy I, 2006. http://tel.archives-ouvertes.fr/tel-00142656.
Full textTrélat, Emmanuel. "Etude asymptotique et transcendance de la fonction valeur en contrôle optimal. Catégorie log-exp en géométrie sous-riemannienne dans le cas Martinet." Dijon, 2000. http://www.theses.fr/2000DIJOS076.
Full textDistexhe, Julie. "Triangulating symplectic manifolds." Doctoral thesis, Universite Libre de Bruxelles, 2019. https://dipot.ulb.ac.be/dspace/bitstream/2013/287522/3/toc.pdf.
Full textIn this thesis, we study symplectic structures in a piecewise linear (PL) setting. The central question is to determine whether a smooth symplectic manifold can be triangulated symplectically, in the sense that there exists a triangulation $h :K -> M$ such that $h^*omega$ is a piecewise constant symplectic form on $K$. We first focus on a simpler related problem, and show that any smooth volume form $Omega$ on $M$ can be triangulated. This means that there always exists a triangulation $h :K -> M$ such that $h^*Omega$ is a piecewise constant volume form. In particular, symplectic surfaces admit symplectic triangulations. Given a closed symplectic manifold $(M,omega)$, we then prove that there exists triangulations $h :K -> M$ for which the piecewise smooth form $h^*omega$ has maximal rank along all the simplices of $K$. This result allows to approximate arbitrarily closely any closed symplectic manifold by a PL one. Finally, we investigate the case of a symplectic submanifold $M$ of an ambient space which is itself symplectically triangulated, and give the construction of a cobordism between $M$ and a piecewise smooth approximation of $M$, triangulated by a symplectic complex.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished
Cortier, Julien. "Étude mathématique de Trous Noirs et de leurs données initiales en Relativité Générale." Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2011. http://tel.archives-ouvertes.fr/tel-00629802.
Full textRichard, Thomas. "Flot de Ricci sans borne supérieure sur la courbure et géométrie de certains espaces métriques." Phd thesis, Université de Grenoble, 2012. http://tel.archives-ouvertes.fr/tel-00768066.
Full textAntonio, Tamarasselvame Nirmal. "Modèle de second gradient adapté aux milieux faiblement continus et mécanique d'Eshelby appliquée à l'indentation du verre." Phd thesis, Université Rennes 1, 2010. http://tel.archives-ouvertes.fr/tel-00557871.
Full textAntonio, Tamarasselvame Nirmal. "Modèle de second gradient adapté aux milieux faiblement continus et mécanique d’Eshelby appliquée à l’indentation du verre." Rennes 1, 2010. https://tel.archives-ouvertes.fr/tel-00557871.
Full textIn a first part, we deal with the so-called weakly continuous media according to an approach based on Riemann-Cartan geometry. We consider a solid body, modelled by a Riemannian manifold, and an Euclidean affine connection, which derives from the metric tensor. The mass density per volume unit may be assumed non constant and some defects, described by discontinuity fields of scalar fields or vectorial fields defined on the manifold, may appear in the body. The inevstigations do not concern the evolution of these fields but take into account their effects on the analysis of the deformation of the body. A possible generalization of this model is to consider an affine connection which deos not derive from the metric induced by the ambiant space. In a such case the torsion tensor and the curvature tensor associated with the affine connection are not necessary null, this corresponds to a second gradient continuum. Both tensors are used to describe the dislocation fields and disclination fields of Volterra. In a second part, we deal with the modelling of the Vickers indentation of glass. We consider a model which uses the schema of inclusion of Eshelby into a semi-infinite matrix, to analyse the stress and displacement fields during the loading process of the indenter. The objective is to determine the densification of the glass beneath the indenter. The semi-analytical results are positively compared with experimental data which are issue from LARMAUR
Butruille, Jean-Baptiste. "Variétés de Gray et géométries spéciales en dimension 6." Phd thesis, Ecole Polytechnique X, 2005. http://tel.archives-ouvertes.fr/tel-00118939.
Full textWang, Jian. "Les 3-variétés contractiles et courbure scalaire positive." Thesis, Université Grenoble Alpes (ComUE), 2019. http://www.theses.fr/2019GREAM076.
Full textThe purposes of this thesis is to understand spaces which carry metrics of positive scalar curvature. There are several topological obstructions for a smooth manifold to have a complete metric of positive scalar curvature. Our goal is to find all obstructions for contractible 3-manifolds and closed 4-manifolds.In dimension 3, we are concerned with the question whether a complete contractible 3-manifold of positive scalar curvature is homeomorphic to mathbb{R} {3}. The topological structure of contractible 3-manifolds could be complicated. For example, the Whitehead manifold is a contractible 3-manifold which is not homeomorphic to bb{R} 3.vspace{2mm}We first prove that the Whitehead manifold does not carry a complete metric of positive scalar curvature. This result can be generalised to the so-called genus one case. Precisely, we show that no contractible genus one 3-manifold admits a complete metric of positive scalar curvature.We then study the fundamental group at infinity, pi{infty} 1, and its relationship with the existence of positive scalar curvature metric. The fundamental group at infinity of a manifold is the inverse limit of the fundamental groups of complements of compact subsets. In this thesis, we give a partial answer to the above question. We prove that a complete contractible 3-manifold with positive scalar curvature and trivial pi {infty} {1} is homeomorphic to mathbb{R} {3}.Finally, we study closed aspherical 4-manifolds. Together with minimal surface theory and the geometrisation conjecture, we show that no closed aspherical 4-manifold with non-trivial first Betti number carries a metric of positive scalar curvature
Kourganoff, Mickaël. "Géométrie et dynamique des espaces de configuration." Thesis, Lyon, École normale supérieure, 2015. http://www.theses.fr/2015ENSL1049/document.
Full textThis thesis is divided into three parts. In the first part, we study linkages (mechanisms made of rigid rods) whose ambiant space is no longer the plane, but various Riemannian manifolds. We study the question of the universality of linkages: this notion corresponds to the idea that every curve would be traced out by a vertex of some linkage, and that any differentiable manifold would be the configuration space of some linkage. We extend universality theorems to the Minkowski plane, the hyperbolic plane, and finally the sphere.Any surface in R^3 can be flattened with respect to the z-axis, and the flattened surface gets close to a billiard table in R^2. In the second part, we show that, under some hypotheses, the geodesic flow of the surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersing, the chaotic properties of the billiard also apply to the geodesic flow: we show that it is Anosov in this case. By applying this result to the theory of linkages, we obtain a new example of Anosov linkage, made of five rods.In the third part, we first consider manifolds with locally metric connections, that is, connections which are locally Levi-Civita connections of Riemannian metrics; we give in this framework an analog of De Rham's decomposition theorem, which usually applies to Riemannian manifolds. In the case such a connection also preserves a conformal structure, we show that this decomposition has at most two factors; moreover, when there are exactly two factors, one of them is the Euclidean space R^q. The proofs of the results of this part use foliations with transverse similarity structures. On these foliations, we give a rigidity theorem of independant interest: they are either transversally flat, or transversally Riemannian
Zang, Yiming. "Les surfaces de Ricci et les surfaces minimales dans les groupes de Lie métriques." Electronic Thesis or Diss., Université de Lorraine, 2022. https://docnum.univ-lorraine.fr/ulprive/DDOC_T_2022_0115_ZANG.pdf.
Full textIn this thesis, we will study some topics related to minimal surfaces in three-dimensional homogeneous manifolds. The first part is devoted to the study of non-positively curved Ricci surfaces with catenodial ends. The idea comes from a famous theorem of Huber. In the first place, we give a definition of catenoidal end for non-positively curved Ricci surfaces with finite total curvature. Secondly, we develop a tool which can be regarded as an analogue of the Weierstrass data. By using this tool, we get some classification results and some non-existence results for non-positively curved Ricci surfaces of genus zero with catenoidal ends. In the end of Chapter 2, we also prove an existence result for non-positively curved Ricci surfaces of arbitrary positive genus with finite many catenoidal ends.In the second part of this thesis, we concern about minimal surfaces in a three-dimensional metric Lie group widetilde{E(2)}, which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. Firstly, a result of Patrangenaru describes the left-invariant metrics as a two-parameter family of metrics. Then we take advantage of a Weierstrass-type representation due to Meeks, Mira, Pérez and Ros to construct a one-parameter family of helicoidal minimal surfaces in widetilde{E(2)} as well as a one-parameter family of minimal annuli which are properly embedded in widetilde{E(2)}. In the end, by a discussion of the limit case of the second family of surfaces, we obtain a new proof of a half-space theorem for minimal surfaces in widetilde{E(2)}
Deschamps, Guillaume. "Espaces twistoriels et structures complexes exotiques." Phd thesis, Université Rennes 1, 2005. http://tel.archives-ouvertes.fr/tel-00011091.
Full textKloeckner, Benoît. "Géométrie des variétés, des espaces de mesures et des espaces de sous-groupes." Habilitation à diriger des recherches, Université de Grenoble, 2012. http://tel.archives-ouvertes.fr/tel-00785679.
Full textCortier, Julien. "Etude mathématique de trous noirs et de leurs données initiales en relativité générale." Thesis, Montpellier 2, 2011. http://www.theses.fr/2011MON20068/document.
Full textThe aim of this thesis is the mathematical study of families of spacetimes satisfying the Einstein's equations of General Relativity. Two methodsare used in this context.The first part, consisting of the first three chapters of this work,investigates the geometric properties of the Emparan-Reall andPomeransky-Senkov families of 5-dimensional spacetimes. We show that they contain a black-hole region, whose event horizon has non-spherical compact cross sections. We construct an analytic extension, and show its maximality and its uniqueness within a natural class in the Emparan-Reallcase. We further establish the Carter-Penrose diagram for these extensions, and analyse the structure of the ergosurface of the Pomeransky-Senkovspacetimes.The second part focuses on the study of initial data, solutions of theconstraint equations induced by the Einstein's equations. We perform agluing construction between a given family of inital data sets andinitial data of Kerr-Kottler-de Sitter spacetimes, using Corvino'smethod.On the other hand, we construct 3-dimensional asymptotically hyperbolicmetrics which satisfy all the assumptions of the positive mass theorem but the completeness, and which display an energy-momentum vector of arbitry causal type
Meyer, Julien. "Quantisation of the Laplacian and a Curved Version of Geometric Quantisation." Doctoral thesis, Universite Libre de Bruxelles, 2016. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/235181.
Full textOption Mathématique du Doctorat en Sciences
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Pillet, Basile. "Géométrie complexe globale et infinitésimale de l'espace des twisteurs d'une variété hyperkählérienne." Thesis, Rennes 1, 2017. http://www.theses.fr/2017REN1S021/document.
Full textThe purpose of this thesis is to construct geometric objects on a manifold C parametrizing rational curves in the twistor space of a hyperkähler manifold. We shall establish a correspondence between the complex geometry of the twistor space and some differential properties of C (differential operators and curvature of a complex riemannian structure inherited from the base hyperkähler manifold). The first chapters gather some classical results of the theory of hyperkähler manifolds and their twistor spaces. In the chapters 4, 5 and 6, we construct an equivalence of categories between bundles on the twistor space which are trivial on each line and bundles with a connexion of C satisfying certain curvature conditions. The chapter 7 extends this correspondence on the cohomological level whereas the chapter 8 explores its infinitesimal version ; it links curvature of the connexion with thickening (in the sense of LeBrun) of the bundle along the lines
Prandi, Dario. "Géométrie et analyse des systèmes de commande avec dérive : planification des mouvements, évolution de la chaleur et de Schrödinger." Phd thesis, Ecole Polytechnique X, 2013. http://pastel.archives-ouvertes.fr/pastel-00878567.
Full textLi, Han-Ping. "L'étude de la règle de métrique riemannienne de Fisher-Rao et des règles de alpha-connexion affine de Chentsov-AmariL'approximation de densité par projection poursuite." Paris 11, 1986. http://www.theses.fr/1986PA112240.
Full textFirst Part. The concept of a Riemannian metric rule and the concept of an affine connexion rule are introduced in a class of statistical experiments. We prove that in the class of regular experiments, the Riemannian metric rule of Fisher-Rao and the α-affine connexion rule of Chentsov-Amari are parameter-free, isomorphism-invariant, embedding-invariant, projectively-invariant and C-continuous. We point out that in the class of discrete experiments; there is a Riemannian metric rule which verifies the isomorphism-invariance and which is not proportional to that of Fisher-Rao. We point out also that in the class of exponential experiments, there is a Riemannian metric rule which verifies embedding-invariant and which is not proportional to that of Fisher-Rao. We give an example to show that the Riemannian metric rule of Fisher-Rao is not continuous in the sens of the Le cam’s deficiency. We prove finally that in the class of separable experiments, all Riemannian metric rules verifying the embedding-invariance and C-continuity are proportional to the Riemannian metric rule of Fisher-Rao and that all affine connexion rules verifying the embedding-invariance and C-continuity are proportional to the α-affine connexion rule of Chentsov-Amari for some real α. Second Part. Certain results on the projection pursuit density approximation are obtained. The procedure (g(m)(x)) mEN for a gaussian density ϕ(µ, Σ) and the speed of convergence are determined. That g(m)(x) situate at the circle joining g(O)(x) and ϕ(µ, Σ) is showed. A comparison of several divergence measures is made
Bour, Vincent. "Flots géométriques d'ordre quatre et pincement intégral de la courbure." Phd thesis, Université de Grenoble, 2012. http://tel.archives-ouvertes.fr/tel-00771720.
Full textKokkonen, Petri. "Étude du modèle des variétés roulantes et de sa commandabilité." Thesis, Paris 11, 2012. http://www.theses.fr/2012PA112317/document.
Full textWe study the controllability of the control system describing the rolling motion, without slipping nor spinning, of two n-dimensional Riemannian manifolds, one against the other.This model is closely related to the concepts of development and holonomy of the manifolds, and it generalizes to the case of affine manifolds.The main contributions are those given in four articles attached to the the thesis.First of them "Rolling manifolds and Controllability: the 3D case"deal with the case where the two manifolds are 3-dimensional. We give the listof all the possible cases for which the system is not controllable.In the second paper "Rolling manifolds on space forms"one of the manifolds is assumed to have constant curvature.We can then reduce the study of controllability to the study of the holonomy groupof a certain vector bundle connection and we show, for example, thatif the manifold with the constant curvature is an n-sphere and ifthis holonomy group does not act transitively,then the other manifold is in fact isometric to the sphere.The third paper "A Characterization of Isometries between Riemannian Manifolds by using Development along Geodesic Triangles"describes, by using the rolling motion (or development) along the loops,an alternative version of the Cartan-Ambrose-Hicks Theorem,which characterizes, among others, the Riemannian isometries.More precisely, we prove that if one starts from a certain initial orientation,and if one only rolls along loops based at the initial point (associated to this orientation),then the two manifolds are isometric if (and only if) the pathstraced by the rolling motion on the other manifolds, are all loops.Finally, the fourth paper "Rolling Manifolds without Spinning"studies the rolling motion, and its controllability, when slipping is allowed.We characterize the structure of all the possible orbits in terms of the holonomy groupsof the manifolds in question. It is also shown that there does not exist anyprincipal bundle structure such that the related distribution becomes a principal distribution,a fact that is to be compared especially to the results of the second article.Furthermore, in the third chapter of the thesis, we construct carefully the rolling modelin the more general framework of affine manifolds, as well as that of Riemannian manifolds,of possibly different dimensions
Kokkonen, Petri. "Etude du modèle des variétés roulantes et de sa commandabilité." Phd thesis, Université Paris Sud - Paris XI, 2012. http://tel.archives-ouvertes.fr/tel-00764158.
Full textThanwerdas, Yann. "Géométries riemanniennes et stratifiées des matrices de covariance et de corrélation." Thesis, Université Côte d'Azur, 2022. http://www.theses.fr/2022COAZ4024.
Full textIn many applications, the data can be represented by covariance matrices or correlation matrices between several signals (EEG, MEG, fMRI), physical quantities (cells, genes), or within a time window (autocorrelation). The set of covariance matrices forms a convex cone that is not a Euclidean space but a stratified space: it has a boundary which is itself a stratified space of lower dimension. The strata are the manifolds of covariance matrices of fixed rank and the main stratum of Symmetric Positive Definite (SPD) matrices is dense in the total space. The set of correlation matrices can be described similarly.Geometric concepts such as geodesics, parallel transport, Fréchet mean were proposed for generalizing classical computations (interpolation, extrapolation, registration) and statistical analyses (mean, principal component analysis, classification, regression) to these non-linear spaces. However, these generalizations rely on the choice of a geometry, that is a basic operator such as a distance, an affine connection, a Riemannian metric, a divergence, which is assumed to be known beforehand. But in practice there is often not a unique natural geometry that satisfies the application constraints. Thus, one should explore more general families of geometries that exploit the data properties.First, the geometry must match the problem. For instance, degenerate matrices must be rejected to infinity whenever covariance matrices must be non-degenerate. Second, we should identify the invariance of the data under natural group transformations: if scaling each variable independently has no impact, then one needs a metric invariant under the positive diagonal group, for instance a product metric that decouples scales and correlations. Third, good numerical properties (closed-form formulae, efficient algorithms) are essential to use the geometry in practice.In my thesis, I study geometries on covariance and correlation matrices following these principles. In particular, I provide the associated geometric operations which are the building blocks for computing with such matrices.On SPD matrices, by analogy with the characterization of affine-invariant metrics, I characterize the continuous metrics invariant by O(n) by means of three multivariate continuous functions. Thus, I build a classification of metrics: the constraints imposed on these functions define nested classes satisfying stability properties. In particular, I reinterpret the class of kernel metrics, I introduce the family of mixed-Euclidean metrics for which I compute the curvature, and I survey and complete the knowledge on the classical metrics (log-Euclidean, Bures-Wasserstein, BKM, power-Euclidean).On full-rank correlation matrices, I compute the Riemannian operations of the quotient-affine metric. Despite its appealing construction and its invariance under permutations, I show that its curvature is of non-constant sign and unbounded from above, which makes this geometry practically very complex. I introduce computationally more convenient Hadamard or even log-Euclidean metrics, along with their geometric operations. To recover the lost invariance under permutations, I define two new permutation-invariant log-Euclidean metrics, one of them being invariant under a natural involution on full-rank correlation matrices. I also provide an efficient algorithm to compute the associated geometric operations based on the scaling of SPD matrices.Finally, I study the stratified Riemannian structure of the Bures-Wasserstein distance on covariance matrices. I compute the domain of definition of geodesics and the injection domain within each stratum and I characterize the length-minimizing curves between all the strata
Grouy, Thibaut. "Radon-type transforms on some symmetric spaces." Doctoral thesis, Universite Libre de Bruxelles, 2019. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/285815.
Full textIn this thesis, we study Radon-type transforms on some symmetric spaces. A Radon-type transform associates to any compactly supported continuous function on a manifold $M$ its integrals over a class $Xi$ of submanifolds of $M$. The problem we address is the inversion of such a transform, that is determining the function in terms of its integrals over the submanifolds in $Xi$. We first present the solution to this inverse problem which is due to Sigurdur Helgason and François Rouvière, amongst others, when $M$ is an isotropic Riemannian symmetric space and $Xi$ a particular orbit of totally geodesic submanifolds of $M$ under the action of a Lie transformation group of $M$. The associated Radon transform is qualified as totally geodesic.On semisimple pseudo-Riemannian symmetric spaces, we consider an other Radon-type transform, which associates to any compactly supported continuous function its orbital integrals, that is its integrals over the orbits of the isotropy subgroup of the transvection group. The inversion of orbital integrals, which is given by a limit-formula, has been obtained by Sigurdur Helgason on Lorentzian symmetric spaces with constant sectional curvature and by Jeremy Orloff on any rank-one semisimple pseudo-Riemannian symmetric space. We solve the inverse problem for orbital integrals on Cahen-Wallach spaces, which are model spaces of solvable indecomposable Lorentzian symmetric spaces.In the last part of the thesis, we are interested in Radon-type transforms on symplectic symmetric spaces with Ricci-type curvature. The inversion of orbital integrals on these spaces when they are semisimple has already been obtained by Jeremy Orloff. However, when these spaces are not semisimple, the orbital integral operator is not invertible. Next, we determine the orbits of symplectic or Lagrangian totally geodesic submanifolds under the action of a Lie transformation group of the starting space. In this context, the technique of inversion that has been developed by Sigurdur Helgason and François Rouvière, amongst others, only works for symplectic totally geodesic Radon transforms on Kählerian symmetric spaces with constant holomorphic curvature. The inversion formulas for these transforms on complex hyperbolic spaces are due to François Rouvière. We compute the inversion formulas for these transforms on complex projective spaces.
Doctorat en Sciences
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Pereira, Mike. "Champs aléatoires généralisés définis sur des variétés riemanniennes : théorie et pratique." Thesis, Paris Sciences et Lettres (ComUE), 2019. http://www.theses.fr/2019PSLEM055.
Full textGeostatistics is the branch of statistics attached to model spatial phenomena through probabilistic models. In particular, the spatial phenomenon is described by a (generally Gaussian) random field, and the observed data are considered as resulting from a particular realization of this random field. To facilitate the modeling and the subsequent geostatistical operations applied to the data, the random field is usually assumed to be stationary, thus meaning that the spatial structure of the data replicates across the domain of study. However, when dealing with complex spatial datasets, this assumption becomes ill-adapted. Indeed, how can the notion of stationarity be defined (and applied) when the data lie on non-Euclidean domains (such as spheres or other smooth surfaces)? Also, what about the case where the data clearly display a spatial structure that varies across the domain? Besides, using more complex models (when it is possible) generally comes at the price of a drastic increase in operational costs (computational and storage-wise), rendering them impossible to apply to large datasets. In this work, we propose a solution to both problems, which relies on the definition of generalized random fields on Riemannian manifolds. On one hand, working with generalized random fields allows to naturally extend ongoing work that is done to leverage a characterization of random fields used in Geostatistics as solutions of stochastic partial differential equations. On the other hand, working on Riemannian manifolds allows to define such fields on both (only) locally Euclidean domains and on locally deformed spaces (thus yielding a framework to account for non-stationary cases). The discretization of these generalized random fields is undertaken using a finite element approach, and we provide an explicit formula for a large class of fields comprising those generally used in applications. Finally, to solve the scalability problem,we propose algorithms inspired from graph signal processing to tackle the simulation, the estimation and the inference of these fields using matrix-free approaches
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.
Full textDoctorat en Sciences
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Dellinger, Marie. "Etude asymptotique et multiplicité pour l'équation de Sobolev Poincaré." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2007. http://tel.archives-ouvertes.fr/tel-00261595.
Full texton considère une edp elliptique non linéaire à exposant critique particulière : l'équation de Sobolev Poincaré. D'une part, nous décrivons le comportement asymptotique d'une suite de solutions de cette équation grâce à une analyse fine de phénomènes de concentration. D'autre part, en imposant des invariances par des groupes d'isométries, nous montrons des résultats de multiplicité de solutions pour cette équation. Notre méthode permet aussi d'obtenir des multiplicités de solutions pour des équations plus classiques provenant du problème deYamabe et de Nirenberg, ainsi que
pour des équations à exposants sur critiques. Notre travail est intimement lié à la description des meilleures constantes dans des inégalités fonctionnelles de Sobolev associées aux équations.
Hafassa, Boutheina. "Deux problèmes de contrôle géométrique : holonomie horizontale et solveur d'esquisse." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS017/document.
Full textWe study two problems arising from geometric control theory. The Problem I consists of extending the concept of horizontal holonomy group for affine manifolds. More precisely, we consider a smooth connected finite-dimensional manifold M, an affine connection ∇ with holonomy group H∇ and ∆ a smooth completely non integrable distribution. We define the ∆-horizontal holonomy group H∆∇ as the subgroup of H∇ obtained by ∇-parallel transporting frames only along loops tangent to ∆. We first set elementary properties of H∆∇ and show how to study it using the rolling formalism. In particular, it is shown that H∆∇ is a Lie group. Moreover, we study an explicit example where M is a free step-two homogeneous Carnot group with m≥2 generators, and ∇ is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that H∆∇ is compact and strictly included in H∇ as soon as m≥3. The Problem II is studying the modeling of the problem of solver sketch. This problem is one of the steps of a CAD/CAM software. Our goal is to achieve a well founded mathematical modeling and systematic the problem of solver sketch. The next step is to understand the convergence of the algorithm, to improve the results and to expand the functionality. The main idea of the algorithm is to replace first the points of the space of spheres by displacements (elements of the group) and then use a Newton's method on Lie groups obtained. In this thesis, we classified the possible displacements of the groups using the theory of Lie groups. In particular, we distinguished three sets, each set containing an object type: the first one is the set of points, denoted Points, the second is the set of lines, denoted Lines, and the third is the set of circles and lines, we note that ∧. For each type of object, we investigated all the possible movements of groups, depending on the desired properties. Finally, we propose to use the following displacement of groups for the displacement of points, the group of translations, which acts transitively on Lines ; for the lines, the group of translations and rotations, which is 3-dimensional and acts transitively (globally but not locally) on Lines ; on lines and circles, the group of anti-translations, rotations and dilations which has dimension 4 and acts transitively (globally but not locally) on ∧
Formont, Pierre. "Outils statistiques et géométriques pour la classification des images SAR polarimétriques hautement texturées." Phd thesis, Université Rennes 1, 2013. http://tel.archives-ouvertes.fr/tel-00983304.
Full textArsigny, Vincent. "Traitement de données dans les groupes de Lie : une approche algébrique. Application au recalage non-linéaire et à l'imagerie du tenseur de diffusion." Phd thesis, Ecole Polytechnique X, 2006. http://tel.archives-ouvertes.fr/tel-00121162.
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