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Auteur : Grafiati
Publié le 10 mars 2023

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1

Falbel, Elisha, Claudio Gorodski et Michel Rumin. « Holonomy of Sub-Riemannian Manifolds ». International Journal of Mathematics 08, no 03 (mai 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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2

Tan, Kang-Hai, et Xiao-Ping Yang. « On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds ». Bulletin of the Australian Mathematical Society 70, no 2 (octobre 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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3

Agrachev, Andrei, Ugo Boscain, Robert Neel et 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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4

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 (juin 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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5

Han, Yanling, Fengyun Fu et Peibiao Zhao. « On semi-symmetric metric connection in sub-Riemannian manifold ». Tamkang Journal of Mathematics 47, no 4 (30 décembre 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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6

Han, Yanling, Fengyun Fu et 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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7

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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8

Rovenski, Vladimir. « Integral Formulas for a Foliation with a Unit Normal Vector Field ». Mathematics 9, no 15 (26 juillet 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 (14 décembre 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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10

DATTA, MAHUYA. « PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD ». International Journal of Mathematics 23, no 02 (février 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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11

Huang, Yong Hong. « Every sub-Riemannian manifold is the Gromov–Hausdorff limit of a sequence Riemannian manifolds ». Acta Mathematica Sinica, English Series 33, no 11 (29 août 2017) : 1565–68. http://dx.doi.org/10.1007/s10114-017-4543-x.

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12

Diniz, M. M., et J. M. M. Veloso. « k-Step Sub-Riemannian Manifold whose Sub-Riemannian Metric Admits a Canonical Extension to a Riemannian Metric ». Journal of Dynamical and Control Systems 16, no 4 (octobre 2010) : 517–38. http://dx.doi.org/10.1007/s10883-010-9105-9.

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13

Barilari, Davide, et Luca Rizzi. « A Formula for Popp’s Volume in Sub-Riemannian Geometry ». Analysis and Geometry in Metric Spaces 1 (14 janvier 2013) : 42–57. http://dx.doi.org/10.2478/agms-2012-0004.

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Abstract For an equiregular sub-Riemannian manifold M, Popp’s volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp’s volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub- Laplacian, namely the one associated with Popp’s volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp’s volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp’s volume is essentially the unique volume with such a property.
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14

Lee, Sungyun. « Volume comparison of Bishop-Gromov type ». Bulletin of the Australian Mathematical Society 45, no 2 (avril 1992) : 241–48. http://dx.doi.org/10.1017/s0004972700030100.

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Bishop-Gromov type comparison theorems for the volume of a tube about a sub-manifold of a complete Riemannian manifold whose Ricci curvature is bounded from below are proved. The Kähler analogue is also proved.
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15

Arguillère, Sylvain, et Emmanuel Trélat. « SUB-RIEMANNIAN STRUCTURES ON GROUPS OF DIFFEOMORPHISMS ». Journal of the Institute of Mathematics of Jussieu 16, no 4 (10 août 2015) : 745–85. http://dx.doi.org/10.1017/s1474748015000249.

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In this paper, we define and study strong right-invariant sub-Riemannian structures on the group of diffeomorphisms of a manifold with bounded geometry. We derive the Hamiltonian geodesic equations for such structures, and we provide examples of normal and of abnormal geodesics in that infinite-dimensional context. The momentum formulation gives a sub-Riemannian version of the Euler–Arnol’d equation. Finally, we establish some approximate and exact reachability properties for diffeomorphisms, and we give some consequences for Moser theorems.
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16

Lučić, Danka, et Enrico Pasqualetto. « Infinitesimal Hilbertianity of Weighted Riemannian Manifolds ». Canadian Mathematical Bulletin 63, no 1 (27 septembre 2019) : 118–40. http://dx.doi.org/10.4153/s0008439519000328.

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AbstractThe main result of this paper is the following: anyweightedRiemannian manifold$(M,g,\unicode[STIX]{x1D707})$,i.e., a Riemannian manifold$(M,g)$endowed with a generic non-negative Radon measure$\unicode[STIX]{x1D707}$, isinfinitesimally Hilbertian, which means that its associated Sobolev space$W^{1,2}(M,g,\unicode[STIX]{x1D707})$is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold$(M,F,\unicode[STIX]{x1D707})$can be isometrically embedded into the space of all measurable sections of the tangent bundle of$M$that are$2$-integrable with respect to$\unicode[STIX]{x1D707}$.By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.
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17

Atceken, Mehmet, SÄuleyman Dirik et ÄUmit Yildirim. « Pseudo-Slant Submanifolds of a Locally Decomposable Riemannian Manifold ». JOURNAL OF ADVANCES IN MATHEMATICS 11, no 8 (20 décembre 2015) : 5587–97. http://dx.doi.org/10.24297/jam.v11i8.1213.

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In this paper, we study pseudo-slant submanifolds of a locally decom- posable Riemannian manifold. We give necessary and suffcient conditions for distributions which are involued in the definition of pseudo-slant sub- manifold to be integrable. We search these type submanifolds with parallel canonical structures and we obtain some new results.
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18

Colin, Ovidiu, Der-Chen Chang et Peter Greiner. « On a Step 2(k + 1) sub-Riemannian manifold ». Journal of Geometric Analysis 14, no 1 (mars 2004) : 1–18. http://dx.doi.org/10.1007/bf02921863.

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19

Ecker, Klaus. « On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetimes ». Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 55, no 1 (août 1993) : 41–59. http://dx.doi.org/10.1017/s1446788700031918.

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AbstractWe prove a priori estimates for the gradient and curvature of spacelike hypersurfaces moving by mean curvature in a Lorentzian manifold. These estimates are obtained under much weaker conditions than have been previously assumed. We also use mean curvature flow in the construction of maximal slices in asymptotically flat spacetimes. An essential tool is a maximum principle for sub-solutions of a parabolic operator on complete Riemannian manifolds with time-dependent metric.
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20

Simić, Slobodan N. « On the sub-Riemannian geometry of contact Anosov flows ». Journal of Topology and Analysis 08, no 01 (23 février 2016) : 187–205. http://dx.doi.org/10.1142/s1793525316500072.

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We investigate certain natural connections between sub-Riemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a sub-Riemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough sub-Riemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.
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21

LEBRUN, CLAUDE. « FANO MANIFOLDS, CONTACT STRUCTURES, AND QUATERNIONIC GEOMETRY ». International Journal of Mathematics 06, no 03 (juin 1995) : 419–37. http://dx.doi.org/10.1142/s0129167x95000146.

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Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M4n, g). If Z also admits a second complex contact structure [Formula: see text], then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).
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22

Galaev, S. V. « Connections with parallel skew-symmetric torsion on sub-Riemannian manifolds ». Differential Geometry of Manifolds of Figures, no 51 (2020) : 49–57. http://dx.doi.org/10.5922/0321-4796-2020-51-7.

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On a sub-Riemannian manifold M of contact type, is considered an N-connection defined by the pair (, N), where is an interior metric connection, is an endomorphism of the distribution D. It is proved that there exists a unique N-connection such that its torsion is skew-symmetric as a contravariant tensor field. A construction of the endomorphism corresponding to such connection is found. The sufficient conditions for the obtained connection to be a metric connec­tion with parallel torsion are given.
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23

Paternain, Gabriel P. « Multiplicity two actions and loop space homology ». Ergodic Theory and Dynamical Systems 13, no 1 (mars 1993) : 143–51. http://dx.doi.org/10.1017/s0143385700007252.

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AbstractWe study Hamiltonian actions of compact Lie groups with low dimensional Marsden-Weinstein reduced spaces. We show that Hamiltonian systems with such symmetry groups have zero topological entropy and easily described dynamics. As a result we show that if the geodesic flow of a compact simply connected Riemannian manifold admits such a symmetry group, then the loop space homology of the manifold grows sub-exponentially with any field coefficient. Topological obstructions for collective complete integrability are thus obtained.
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24

Lerario, A., et L. Rizzi. « How many geodesics join two points on a contact sub-Riemannian manifold ? » Journal of Symplectic Geometry 15, no 1 (2017) : 247–305. http://dx.doi.org/10.4310/jsg.2017.v15.n1.a7.

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25

Williams, Simon, Arthur George Suvorov, Zengfu Wang et Bill Moran. « The Information Geometry of Sensor Configuration ». Sensors 21, no 16 (4 août 2021) : 5265. http://dx.doi.org/10.3390/s21165265.

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In problems of parameter estimation from sensor data, the Fisher information provides a measure of the performance of the sensor; effectively, in an infinitesimal sense, how much information about the parameters can be obtained from the measurements. From the geometric viewpoint, it is a Riemannian metric on the manifold of parameters of the observed system. In this paper, we consider the case of parameterized sensors and answer the question, “How best to reconfigure a sensor (vary the parameters of the sensor) to optimize the information collected?” A change in the sensor parameters results in a corresponding change to the metric. We show that the change in information due to reconfiguration exactly corresponds to the natural metric on the infinite-dimensional space of Riemannian metrics on the parameter manifold, restricted to finite-dimensional sub-manifold determined by the sensor parameters. The distance measure on this configuration manifold is shown to provide optimal, dynamic sensor reconfiguration based on an information criterion. Geodesics on the configuration manifold are shown to optimize the information gain but only if the change is made at a certain rate. An example of configuring two bearings-only sensors to optimally locate a target is developed in detail to illustrate the mathematical machinery, with Fast Marching methods employed to efficiently calculate the geodesics and illustrate the practicality of using this approach.
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Barilari, Davide, Ugo Boscain et Daniele Cannarsa. « On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds ». ESAIM : Control, Optimisation and Calculus of Variations 28 (2022) : 9. http://dx.doi.org/10.1051/cocv/2021104.

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Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.
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27

SCHULTE-HERBRÜGGEN, THOMAS, STEFFEN J. GLASER, GUNTHER DIRR et UWE HELMKE. « GRADIENT FLOWS FOR OPTIMIZATION IN QUANTUM INFORMATION AND QUANTUM DYNAMICS : FOUNDATIONS AND APPLICATIONS ». Reviews in Mathematical Physics 22, no 06 (juillet 2010) : 597–667. http://dx.doi.org/10.1142/s0129055x10004053.

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Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is sufficiently general for setting up gradient flows on (sub)manifolds, Lie (sub)groups, and (reductive) homogeneous spaces. Relevant convergence conditions are discussed, in particular for gradient flows on compact and analytic manifolds. This is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SU loc (2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SU loc (2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.
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Agostiniani, Virginia, Lorenzo Mazzieri et Francesca Oronzio. « A geometric capacitary inequality for sub-static manifolds with harmonic potentials ». Mathematics in Engineering 4, no 2 (2021) : 1–40. http://dx.doi.org/10.3934/mine.2022013.

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<abstract><p>In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $ \{F_{\beta}\} $ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $ \beta = \frac{n-2}{n-1} $ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.</p></abstract>
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Plaszczyk, Mariusz. « The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds ». Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 69, no 1 (30 novembre 2015) : 91. http://dx.doi.org/10.17951/a.2015.69.1.91.

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If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one J<sup>r</sup>TM → J<sup>r</sup>T*M between the r-th order prolongation J<sup>r</sup>TM of tangent TM and the r-th order prolongation J<sup>r</sup>T*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps D<sub>M</sub>(g) : J<sup>r</sup>TM → J<sup>r</sup>T*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.
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Guan, Jianyun, et Haiming Liu. « The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric ». Journal of Function Spaces 2021 (8 août 2021) : 1–14. http://dx.doi.org/10.1155/2021/1431082.

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The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1 , 1 , g λ 1 , λ 2 , where λ 1 ≥ λ 2 > 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold Sol 3 = E 1 , 1 , g 1 , 1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in E 1 , 1 , g L λ 1 , λ 2 away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.
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31

Wu, Feifan, et Wei Wang. « The Bochner-type formula and the first eigenvalue of the sub-Laplacian on a contact Riemannian manifold ». Differential Geometry and its Applications 37 (décembre 2014) : 66–88. http://dx.doi.org/10.1016/j.difgeo.2014.10.002.

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Le Donne, Enrico, et Séverine Rigot. « Besicovitch Covering Property on graded groups and applications to measure differentiation ». Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no 750 (1 mai 2019) : 241–97. http://dx.doi.org/10.1515/crelle-2016-0051.

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Abstract We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.
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33

Bernicot, Frédéric, et Yannick Sire. « Propagation of low regularity for solutions of nonlinear PDEs on a Riemannian manifold with a sub-Laplacian structure ». Annales de l'Institut Henri Poincare (C) Non Linear Analysis 30, no 5 (septembre 2013) : 935–58. http://dx.doi.org/10.1016/j.anihpc.2012.12.005.

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34

Košťáková, Petra, et Pavel Stovicek. « THE AHARONOV-BOHM HAMILTONIAN WITH TWO VORTICES REVISITED ». Acta Polytechnica 56, no 3 (30 juin 2016) : 224. http://dx.doi.org/10.14311/ap.2016.56.0224.

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<p>We consider an invariant quantum Hamiltonian <em>H</em> = −Δ<em><sub>LB</sub></em> + <em>V</em> in the <em>L</em><sup>2</sup> space based on a Riemannian manifold <em>˜M</em> with a discrete symmetry group Γ. To any unitary representation Λ of Γ one can relate another operator on <em>M</em> = <em>˜M</em> /Γ, called <em>H</em><sub>Λ</sub>, which formally corresponds to the same differential operator as <em>H</em> but which is determined by quasi-periodic boundary conditions. As originally observed by Schulman in theoretical physics and Sunada in mathematics, one can construct the propagator associated with <em>H</em><sub>Λ</sub> provided one knows the propagator associated with <em>H</em>. This approach is reviewed and demonstrated on a quantum model describing a charged particle on the plane with two Aharonov-Bohm vortices. The construction of the propagator is explained in full detail including all substantial intermediate steps.</p>
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35

Piccione, Paolo, et Daniel V. Tausk. « Lagrangian and Hamiltonian formalism for constrained variational problems ». Proceedings of the Royal Society of Edinburgh : Section A Mathematics 132, no 6 (décembre 2002) : 1417–37. http://dx.doi.org/10.1017/s0308210500002183.

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We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ TM on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set ΩPQ(M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M. Under suitable assumptions, the set ΩPQ(M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on TM* using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive-definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [8]). By adding a potential energy term to L, we obtain again the equations of motion for the Vakonomic mechanics with non-holonomic constraints (see [6]).
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36

Reich, M., et C. Heipke. « A global approach for image orientation using Lie algebraic rotation averaging and convex L&infin ; minimisation ». ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-3 (11 août 2014) : 265–72. http://dx.doi.org/10.5194/isprsarchives-xl-3-265-2014.

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In this paper we present a new global image orientation approach for a set of multiple overlapping images with given homologous point tuples which is based on a two-step procedure. The approach is independent on initial values, robust with respect to outliers and yields the global minimum solution under relatively mild constraints. The first step of the approach consists of the estimation of global rotation parameters by averaging relative rotation estimates for image pairs (these are determined from the homologous points via the essential matrix in a pre-processing step). For the averaging we make use of algebraic group theory in which rotations, as part of the special orthogonal group <i>SO(3)</i>, form a Lie group with a Riemannian manifold structure. This allows for a mapping to the local Euclidean tangent space of <i>SO(3)</i>, the Lie algebra. In this space the redundancy of relative orientations is used to compute an average of the absolute rotation for each image and furthermore to detect and eliminate outliers. In the second step translation parameters and the object coordinates of the homologous points are estimated within a convex <i>L</i><sub>&infin;</sub> optimisation, in which the rotation parameters are kept fixed. As an optional third step the results can be used as initial values for a final bundle adjustment that does not suffer from bad initialisation and quickly converges to a globally optimal solution. We investigate our approach for global image orientation based on synthetic data. The results are compared to a robust least squares bundle adjustment. In this way we show that our approach is independent of initial values and more robust against outliers than a conventional bundle adjustment.
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37

Asselmeyer-Maluga, Torsten. « Quantum computing and the brain : quantum nets, dessins d’enfants and neural networks ». EPJ Web of Conferences 198 (2019) : 00014. http://dx.doi.org/10.1051/epjconf/201919800014.

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In this paper, we will discuss a formal link between neural networks and quantum computing. For that purpose we will present a simple model for the description of the neural network by forming sub-graphs of the whole network with the same or a similar state. We will describe the interaction between these areas by closed loops, the feedback loops. The change of the graph is given by the deformations of the loops. This fact can be mathematically formalized by the fundamental group of the graph. Furthermore the neuron has two basic states |0〉 (ground state) and |1〉 (excited state). The whole state of an area of neurons is the linear combination of the two basic state with complex coefficients representing the signals (with 3 Parameters: amplitude, frequency and phase) along the neurons. If something changed in this area, we need a transformation which will preserve this general form of a state (mathematically, this transformation must be an element of the group S L(2; C)). The same argumentation must be true for the feedback loops, i.e. a general transformation of states along the feedback loops is an assignment of this loop to an element of the transformation group. Then it can be shown that the set of all signals forms a manifold (character variety) and all properties of the network must be encoded in this manifold. In the paper, we will discuss how to interpret learning and intuition in this model. Using the Morgan-Shalen compactification, the limit for signals with large amplitude can be analyzed by using quasi-Fuchsian groups as represented by dessins d’enfants (graphs to analyze Riemannian surfaces). As shown by Planat and collaborators, these dessins d’enfants are a direct bridge to (topological) quantum computing with permutation groups. The normalization of the signal reduces to the group S U(2) and the whole model to a quantum network. Then we have a direct connection to quantum circuits. This network can be transformed into operations on tensor networks. Formally we will obtain a link between machine learning and Quantum computing.
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Gordina, Maria, et Thomas Laetsch. « Sub-Laplacians on Sub-Riemannian Manifolds ». Potential Analysis 44, no 4 (22 février 2016) : 811–37. http://dx.doi.org/10.1007/s11118-016-9532-7.

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39

Vodop’yanov, S. K., et I. G. Markina. « Classification of sub-Riemannian manifolds ». Siberian Mathematical Journal 39, no 6 (décembre 1998) : 1096–111. http://dx.doi.org/10.1007/bf02674121.

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40

Rovenski, Vladimir. « Integral Formulas for Almost Product Manifolds and Foliations ». Mathematics 10, no 19 (5 octobre 2022) : 3645. http://dx.doi.org/10.3390/math10193645.

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Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with k=2,3 distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds.
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41

Autenried, Christian, et Irina Markina. « Sub-Riemannian Geometry of Stiefel Manifolds ». SIAM Journal on Control and Optimization 52, no 2 (janvier 2014) : 939–59. http://dx.doi.org/10.1137/130922537.

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42

Fässler, Katrin, Anton Lukyanenko et Kirsi Peltonen. « Quasiregular Mappings on Sub-Riemannian Manifolds ». Journal of Geometric Analysis 26, no 3 (21 août 2015) : 1754–94. http://dx.doi.org/10.1007/s12220-015-9607-5.

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43

Berestovskii, Valerii N. « Curvatures of homogeneous sub-Riemannian manifolds ». European Journal of Mathematics 3, no 4 (31 juillet 2017) : 788–807. http://dx.doi.org/10.1007/s40879-017-0171-3.

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Figalli, Alessio, et Ludovic Rifford. « Mass Transportation on Sub-Riemannian Manifolds ». Geometric and Functional Analysis 20, no 1 (23 février 2010) : 124–59. http://dx.doi.org/10.1007/s00039-010-0053-z.

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45

Karmanova, M. B. « Local metric properties of level surfaces on Carnot–Caratheodory spaces ». Доклады Академии наук 484, no 5 (16 mai 2019) : 527–31. http://dx.doi.org/10.31857/s0869-56524845527-531.

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For level sets of -mappings of Carnot manifolds to Carnot–Caratheodory spaces, we introduce an adequate local metric characteristic. Moreover, for mappings defined on Carnot groups, we construct a special adapted basis in the preimage such that it assigns a suitable local sub-Riemannian structure on a complement of a kernel of a sub-Riemannian differential to the initial sub-Riemannian structure in the image.
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46

Rizzi, Luca, et Pavel Silveira. « SUB-RIEMANNIAN RICCI CURVATURES AND UNIVERSAL DIAMETER BOUNDS FOR 3-SASAKIAN MANIFOLDS ». Journal of the Institute of Mathematics of Jussieu 18, no 4 (21 juin 2017) : 783–827. http://dx.doi.org/10.1017/s1474748017000226.

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For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev–Zelenko–Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet–Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\unicode[STIX]{x1D70B}$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations: $$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$ whose exact sub-Riemannian diameter is $\unicode[STIX]{x1D70B}$, for all $d\geqslant 1$.
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Baldi, Annalisa, Maria Carla Tesi et Francesca Tripaldi. « Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry ». Advanced Nonlinear Studies 22, no 1 (1 janvier 2022) : 484–516. http://dx.doi.org/10.1515/ans-2022-0022.

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Abstract In this article, we establish a Gaffney type inequality, in W ℓ , p {W}^{\ell ,p} -Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p ∈ ] 1 , ∞ [ p\in ]1,\infty {[} and ℓ = 1 , 2 \ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.
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Le Donne, Enrico, Gian Paolo Leonardi, Roberto Monti et Davide Vittone. « Corners in non-equiregular sub-Riemannian manifolds ». ESAIM : Control, Optimisation and Calculus of Variations 21, no 3 (1 mai 2015) : 625–34. http://dx.doi.org/10.1051/cocv/2014041.

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Baudoin, Fabrice, et Michel Bonnefont. « Sub-Riemannian balls in CR Sasakian manifolds ». Proceedings of the American Mathematical Society 141, no 11 (24 juillet 2013) : 3919–24. http://dx.doi.org/10.1090/s0002-9939-2013-11783-1.

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Hladky, Robert K. « Intrinsic complements of equiregular sub-Riemannian manifolds ». Geometriae Dedicata 173, no 1 (26 octobre 2013) : 89–103. http://dx.doi.org/10.1007/s10711-013-9930-6.

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