Relevant bibliographies by topics / Sub-Riemannian manifold

Academic literature on the topic 'Sub-Riemannian manifold'

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Published: 10 March 2023

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Journal articles on the topic "Sub-Riemannian manifold"

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Falbel, Elisha, Claudio Gorodski, and Michel Rumin. "Holonomy of Sub-Riemannian Manifolds." International Journal of Mathematics 08, no. 03 (May 1997): 317–44. http://dx.doi.org/10.1142/s0129167x97000159.

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A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric. We study the holonomy and the horizontal holonomy (i.e. holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection. In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (i.e. homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry).
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Tan, Kang-Hai, and Xiao-Ping Yang. "On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds." Bulletin of the Australian Mathematical Society 70, no. 2 (October 2004): 177–98. http://dx.doi.org/10.1017/s0004972700034407.

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We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the sub-Riemannian structure and is “symmetric” and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.
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Agrachev, Andrei, Ugo Boscain, Robert Neel, and Luca Rizzi. "Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 3 (2018): 1075–105. http://dx.doi.org/10.1051/cocv/2017037.

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We relate some constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.
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BEJANCU, AUREL. "A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (I)." International Journal of Geometric Methods in Modern Physics 08, no. 04 (June 2011): 725–52. http://dx.doi.org/10.1142/s0219887811005361.

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We study the geometry of a sub-Riemannian manifold (M, HM, VM, g), where HM and VM are the horizontal and vertical distribution respectively, and g is a Riemannian extension of the Riemannian metric on HM. First, without the assumption that HM and VM are orthogonal, we construct a sub- Riemannian connection ▽ on HM and prove some Bianchi identities for ▽. Then, we introduce the horizontal sectional curvature, prove a Schur theorem for sub-Riemannian geometry and find a class of sub-Riemannian manifolds of constant horizontal curvature. Finally, we define the horizontal Ricci tensor and scalar curvature, and some sub-Riemannian differential operators (gradient, divergence, Laplacian), extending some results from geometry to the sub-Riemannian setting.
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Han, Yanling, Fengyun Fu, and Peibiao Zhao. "On semi-symmetric metric connection in sub-Riemannian manifold." Tamkang Journal of Mathematics 47, no. 4 (December 30, 2016): 373–84. http://dx.doi.org/10.5556/j.tkjm.47.2016.1908.

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The authors firstly in this paper define a semi-symmetric metric non-holonomic connection (in briefly, SS-connection) on sub-Riemannian manifolds. An invariant under a SS-connection transformation is obtained. The authors then further give a result that a sub-Riemannian manifold $(M,V_{0},g,\bar{\nabla})$ is locally horizontally flat if and only if $M$ is horizontally conformally flat and horizontally Ricci flat.
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Han, Yanling, Fengyun Fu, and Peibiao Zhao. "A class of non-holonomic projective connections on sub-Riemannian manifolds." Filomat 31, no. 5 (2017): 1295–303. http://dx.doi.org/10.2298/fil1705295h.

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The authors define a semi-symmetric non-holonomic (SSNH)-projective connection on sub-Riemannian manifolds and find an invariant of the SSNH-projective transformation. The authors further derive that a sub-Riemannian manifold is of projective flat if and only if the Schouten curvature tensor of a special SSNH-connection is zero.
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Galaev, S. "On geometry of sub-Riemannian η-Einstein manifolds." Differential Geometry of Manifolds of Figures, no. 50 (2018): 68–81. http://dx.doi.org/10.5922/10.5922/0321-4796-2019-50-9.

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On a sub-Riemannian manifold of contact type a connection  with torsion is considered, called in the work a Ψ-connection. A Ψ- connection is a particular case of an N-connection. On a sub-Riemannian manifold, a Ψ-connection is defined up to an endomorphism  :DD of a distribution D, this endomorphism is called in the work the structure endomorphism. The endomorphism ψ is uniquely defined by the following relations:  0,   (x, y)  g( x, y), x, yD. If the distribution of a sub-Riemannian manifold is integrable, then the Ψ-connection is of the class of the quarter-symmetric connections. It is proved that the Ψ- connection is a metric connection if and only if the structure vector field of the sub-Riemannian structure is integrable. A formula expressing the Ψ-connections in terms of the Levi-Civita connection of the sub- Riemannian manifold is obtained. The components of the curvature tensors and the Ricci-tensors of the Ψ-connection and of the Levi-Civita connection are computed. It is proved that if a sub-Riemannian manifold is an η-Einstein manifold, then it is also an η-Einstein manifold with respect to the Ψ-connection. The converse holds true only under the condition that the trace of the structure endomorphism Ψ is a constant not depending on a point of the manifold. The paper is completed by the theorem stating that a Sasaki manifold is an η-Einstein manifold if and only if M is an η-Einstein manifold with respect to the Ψ-connection.
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Rovenski, Vladimir. "Integral Formulas for a Foliation with a Unit Normal Vector Field." Mathematics 9, no. 15 (July 26, 2021): 1764. http://dx.doi.org/10.3390/math9151764.

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In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D=TF⊕span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.
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Rovenski, Vladimir. "Geometric Inequalities for a Submanifold Equipped with Distributions." Mathematics 10, no. 24 (December 14, 2022): 4741. http://dx.doi.org/10.3390/math10244741.

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The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions.
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DATTA, MAHUYA. "PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD." International Journal of Mathematics 23, no. 02 (February 2012): 1250043. http://dx.doi.org/10.1142/s0129167x12500437.

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In this article, we obtain the following generalization of isometric C1-immersion theorem of Nash and Kuiper. Let M be a smooth manifold of dimension m and H a rank k subbundle of the tangent bundle TM with a Riemannian metric gH. Then the pair (H, gH) defines a sub-Riemannian structure on M. We call a C1-map f : (M, H, gH) → (N, h) into a Riemannian manifold (N, h) a partial isometry if the derivative map df restricted to H is isometric, that is if f*h|H = gH. We prove that if f0 : M → N is a smooth map such that df0|H is a bundle monomorphism and [Formula: see text], then f0 can be homotoped to a C1-map f : M → N which is a partial isometry, provided dim N > k. As a consequence of this result, we obtain that every sub-Riemannian manifold (M, H, gH) admits a partial isometry in ℝn, provided n ≥ m + k.
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Dissertations / Theses on the topic "Sub-Riemannian manifold"

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Giovannardi, Gianmarco. "Geometric properties of 2-dimensional minimal surfaces in a sub-Riemannian manifold which models the Visual Cortex." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/11473/.

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In this paper we study the notion of degree forsubmanifolds embedded in an equiregular sub-Riemannian manifold and we provide the definition of their associated area functional. In this setting we prove that the Hausdorff dimension of a submanifold coincides with its degree, as stated by Gromov. Using these general definitions we compute the first variation for surfaces embedded in low dimensional manifolds and we obtain the partial differential equation associated to minimal surfaces. These minimal surfaces have several applications in the neurogeometry of the visual cortex.
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Whiting, James K. (James Kalani) 1980. "Path optimization using sub-Riemannian manifolds with applications to astrodynamics." Thesis, Massachusetts Institute of Technology, 2011. http://hdl.handle.net/1721.1/63035.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 131).
Differential geometry provides mechanisms for finding shortest paths in metric spaces. This work describes a procedure for creating a metric space from a path optimization problem description so that the formalism of differential geometry can be applied to find the optimal paths. Most path optimization problems will generate a sub-Riemannian manifold. This work describes an algorithm which approximates a sub-Riemannian manifold as a Riemannian manifold using a penalty metric so that Riemannian geodesic solvers can be used to find the solutions to the path optimization problem. This new method for solving path optimization problems shows promise to be faster than other methods, in part because it can easily run on parallel processing units. It also provides some geometrical insights into path optimization problems which could provide a new way to categorize path optimization problems. Some simple path optimization problems are described to provide an understandable example of how the method works and an application to astrodynamics is also given.
by James K. Whiting.
Ph.D.
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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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Wang, Jing. "Sub-Riemannian heat kernels on model spaces and curvature-dimension inequalities on contact manifolds." Thesis, Purdue University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3636683.

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This dissertation contains two research directions. In the first direction, we deduce explicit expressions of the subelliptic heat kernels on three sub-Riemannian model spaces: the Cauchy-Riemann sphere, the anti-de Sitter space and the Quaternionic sphere. From these explicit subelliptic heat kernels we then derive several by products: the Green function of the conformal sub-Laplacian, the small-time estimates of the subel- liptic heat kernels, and the sub-Riemannian distance. The key point is to work in cylindrical coordinates that reflect the symmetries coming from the Hopf fibration of these model spaces. In the second direction we study the extension of the Baudoin-Garofalo type curvature dimension inequality from the sub-Riemannian transversal symmetric setting to any contact manifold. In particular, the Sasakian condition is no longer assumed which leads to the appearance of new strongly non-linear term in the curvature dimension inequality. This new curvature dimension condition is then used to study several interesting aspects in geometry and analysis: The stochastic completeness of the heat semigroup, geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results), gradient bounds for the heat semigroup, and spectral gap estimates for the sub-Laplacian.

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Kokkonen, Petri. "Étude du modèle des variétés roulantes et de sa commandabilité." Thesis, Paris 11, 2012. http://www.theses.fr/2012PA112317/document.

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Nous étudions la commandabilité du système de contrôle décrivant le procédé de roulement, sans glissement ni pivotement, de deux variétés riemanniennes n-dimensionnelles, l'une sur l'autre. Ce modèle est étroitement associé aux concepts de développement et d'holonomie des variétés, et il se généralise au cas de deux variétés affines. Les contributions principales sont celles données dans quatre articles, attachés à la fin de la thèse.Le premier d'entre eux «Rolling manifolds and Controllability : the 3D case»traite le cas où les deux variétés sont 3-dimensionelles. Nous donnons alors, la liste des cas possibles pour lesquelles le système n'est pas commandable.Dans le deuxième papier «Rolling manifolds on space forms», l'une des deux variétés est supposée être de courbure constante. On peut alors réduire l'étude de commandabilité à l'étude du groupe d'holonomie d'une certaine connexion vectorielle et on démontre, par exemple, que si la variété à courbure constante est une sphère n-dimensionelle et si ce groupe de l'holonomie n'agit pas transitivement, alors l'autre variété est en fait isométrique à la sphère.Le troisième article «A Characterization of Isometries between Riemannian Manifolds by using Development along Geodesic Triangles» décrit, en utilisant le procédé de roulement (ou développement) le long des lacets, une version alternative du théorème de Cartan-Ambrose-Hicks, qui caractérise, entre autres, les isométries riemanniennes. Plus précisément, on prouve que si on part d'une certaine orientation initiale, et si on ne roule que le long des lacets basés au point initial (associé à cette orientation), alors les deux variétés sont isométriques si (et seulement si) les chemins tracés par le procédé de roulement sur l'autre variété, sont tous des lacets.Finalement, le quatrième article «Rolling Manifolds without Spinning» étudie le procédé de roulement et sa commandabilité dans le cas où l'on ne peut pas pivoter. On caractérise alors les structures de toutes les orbites possibles en termes des groupes d'holonomie des variétés en question. On montre aussi qu'il n'existe aucune structure de fibré principal sur l'espace d'état tel que la distribution associée à ce modèle devienne une distribution principale, ce qui est à comparer notamment aux résultats du deuxième article.Par ailleurs, dans la troisième partie de cette thèse, nous construisons soigneusement le modèle de roulement dans le cadre plus général des variétés affines, ainsi que dans celui des variétés riemanniennes de dimensiondifférente
We 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
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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.

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Nous étudions deux problèmes différents qui ont leur origine dans la théorie du contrôle géométrique. Le Problème I consiste à étendre le concept du groupe d'holonomie horizontale sur une variété affine. Plus précisément, nous considérons une variété connexe lisse de dimension finie M, une connexion affine ∇ avec le groupe d'holonomie H∇ et une distribution lisse ∆ complètement non intégrable. Dans un premier temps, nous définissons le groupe d'holonomie ∆-horizontale H∆∇ comme le sous-groupe de H∇ obtenu par le transport parallèle le long des lacets tangents à ∆. Nous donnons les propriétés élémentaires de H∆∇ et ensuite nous faisons une étude détaillée en utilisant le formalisme de roulement. Il est montré en particulier que H∆∇ est un groupe de Lie. Dans un second temps, nous avons étudié un exemple explicite où M est un groupe de Carnot libre d'ordre 2 avec m ≥ 2 générateurs, et ∇ est la connexion de Levi-Civita associé à une métrique riemannienne sur M. Nous avons montré dans ce cas particulier que H∆∇ est compact et strictement inclus dans H∇ dès que m≥3. Le Problème II étudie la modélisation du problème du solveur d'esquisse. Ce problème est une des étapes d'un logiciel de CFAO. Notre but est d'arriver à une modélisation mathématique bien fondée et systématique du problème du solveur d'esquisse. Il s'agira ensuite de comprendre la convergence de l'algorithme, d'en améliorer les résultats et d'en étendre les fonctionnalités. L'idée directrice de l'algorithme est de remplacer tout d'abord les points de l'espace des sphères par des déplacements (éléments du groupe) et puis d'utiliser une méthode de Newton sur les groupes de Lie ainsi obtenus. Dans cette thèse, nous avons classifié les groupes de déplacements possibles en utilisant la théorie des groupes de Lie. En particulier, nous avons distingué trois ensembles, chaque ensemble contenant un type d'objet: le premier est l'ensemble des points, noté Points , le deuxième est l'ensemble des droites, noté Droites, et le troisième est l'ensemble des cercles et des droites, que nous notons ∧. Pour chaque type d'objet nous avons étudié tous les groupes de déplacements possibles, selon les propriétés souhaitées. Nous proposons finalement d'utiliser les groupes de déplacements suivant: pour le déplacement des points, le groupe des translations, qui agit transitivement sur Points ; pour les droites, le groupe des translations et rotations, qui est de dimension 3 et agit transitivement (globalement mais pas localement) sur Droites ; sur les droites et cercles, le groupe des anti-translations, rotations et dilatations qui est de dimension 4 et agit transitivement (globalement mais pas localement) sur ∧
We 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 ∧
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Giovannardi, Gianmarco. "Variations for submanifolds of fixed degree." Doctoral thesis, 2020. https://hdl.handle.net/2158/1287865.

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The aim of this PhD thesis is to study the area functional for submanifolds immersed in an equiregular graded manifold. This setting, extends the sub-Riemannian one, removing the bracket generating condition. However, even in the sub-Riemannian setting only sub-manifolds of dimension or codimension one have been extensively studied. We will study the general case and observe that in higher codimension new phenomena arise, which can not show up in the Riemannian case. In particular, we will prove the existence of isolated surfaces, which do not admit degree preserving variation: a phenomena observed by now only for curves, related to the notion of abnormal geodesics.
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Subils, Mauro. "El problema de equivalencia y la prolongación de Tanaka para distribuciones totalmente no integrables." Doctoral thesis, 2015. http://hdl.handle.net/11086/2789.

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Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física, 2015.
Esta tesis trata del Problema de Equivalencia para distribuciones y estructuras geométricas asociadas, o sub-estructuras. Repasamos el método de equivalencia de Cartan y lo comparamos con el Método de Prolongación de Tanaka, que refina ese para estructuras asociadas a distribuciones. Damos una breve introducción a las conexiones de Cartan, que es lo se aspira obtener al resolver un Problema de Equivalencia, y obtenemos algunos resultados sobre su existencia y unicidad. Finalmente, aplicamos la prolongación de Tanaka geométrica a ejemplos distinguidos de distribuciones con métricas subriemannianas y subconformes, obteniendo sus conexiones de Cartan normales y sus invariantes fundamentales.
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Books on the topic "Sub-Riemannian manifold"

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Barilari, Davide, Ugo Boscain, and Mario Sigalotti, eds. Geometry, Analysis and Dynamics on sub-Riemannian Manifolds. Zuerich, Switzerland: European Mathematical Society Publishing House, 2016. http://dx.doi.org/10.4171/162.

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Barilari, Davide, Ugo Boscain, and Mario Sigalotti, eds. Geometry, Analysis and Dynamics on sub-Riemannian Manifolds. Zuerich, Switzerland: European Mathematical Society Publishing House, 2016. http://dx.doi.org/10.4171/163.

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E, Chang Der-chen, ed. Sub-Riemannian geometry: General theory and examples. Cambridge: Cambridge University Press, 2009.

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Calin, Ovidiu, and Der-Chen Chang. Sub-Riemannian Geometry: General Theory and Examples. Cambridge University Press, 2013.

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Calin, Ovidiu, and Der-Chen Chang. Sub-Riemannian Geometry: General Theory and Examples. Cambridge University Press, 2013.

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Calin, Ovidiu, and Der-Chen Chang. Sub-Riemannian Geometry: General Theory and Examples. Cambridge University Press, 2009.

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Gauthier, Jean-Paul, Ugo Boscain, Andrey Sarychev, Gianna Stefani, and Mario Sigalotti. Geometric Control Theory and Sub-Riemannian Geometry. Springer London, Limited, 2014.

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Geometric Control Theory and Sub-Riemannian Geometry. Springer, 2014.

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Gauthier, Jean-Paul, Ugo Boscain, Andrey Sarychev, Gianna Stefani, and Mario Sigalotti. Geometric Control Theory and Sub-Riemannian Geometry. Springer International Publishing AG, 2016.

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Danielli, Donatella, Luca Capogna, Scott D. Pauls, and Jeremy Tyson. Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Birkhauser Verlag, 2007.

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Book chapters on the topic "Sub-Riemannian manifold"

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Sverdlov, Roman, and Dimiter Vassilev. "On Sub-Riemannian and Riemannian Spaces Associated to a Lorentzian Manifold." In Trends in Mathematics, 503–11. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-87502-2_51.

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Markina, Irina, and Stephan Wojtowytsch. "On the Alexandrov Topology of sub-Lorentzian Manifolds." In Geometric Control Theory and Sub-Riemannian Geometry, 287–311. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-02132-4_17.

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Ghezzi, Roberta, and Frédéric Jean. "Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds." In Geometric Control Theory and Sub-Riemannian Geometry, 201–18. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-02132-4_13.

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Baudoin, Fabrice. "Geometric Inequalities on Riemannian and Sub-Riemannian Manifolds by Heat Semigroups Techniques." In Lecture Notes in Mathematics, 7–91. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-84141-6_2.

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Pesenson, Isaac. "Parseval Space-Frequency Localized Frames on Sub-Riemannian Compact Homogeneous Manifolds." In Frames and Other Bases in Abstract and Function Spaces, 413–33. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-55550-8_17.

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Atçeken, Mehmet, Ümit Yildirim, and Süleyman Dirik. "Sub-Manifolds of a Riemannian Manifold." In Manifolds - Current Research Areas. InTech, 2017. http://dx.doi.org/10.5772/65948.

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"Sub-Riemannian Geometry, Heisenberg Manifolds and Quantum Mechanics of Landau Levels." In Applications of Contact Geometry and Topology in Physics, 99–130. WORLD SCIENTIFIC, 2013. http://dx.doi.org/10.1142/9789814412094_0006.

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Conference papers on the topic "Sub-Riemannian manifold"

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Liu, Sidun, Peng Qiao, Yong Dou, and Ruochun Jin. "Searching Latent Sub-Goals in Hierarchical Reinforcement Learning as Riemannian Manifold Optimization." In 2022 IEEE International Conference on Multimedia and Expo (ICME). IEEE, 2022. http://dx.doi.org/10.1109/icme52920.2022.9859878.

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You might also be interested in the extended bibliographies on the topic 'Sub-Riemannian manifold' for particular source types:
Journal articles
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