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The tangent space in sub-Riemannian geometry
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Tangent spaces of a sub-Riemannian manifold are themselves sub-riemannians as mentioned in this paper, and they come with an algebraic structure: nilpotent Lie groups with dilations.Abstract:
Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G.read more
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Sobolev met Poincaré
Piotr Hajłasz,Pekka Koskela +1 more
TL;DR: In this article, the Poincare and Sobolev inequalities, pointwise estimates, and pointwise classifications of Soboleve classes are discussed. But they do not cover the necessary conditions for Poincarse inequalities.
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Carnot-Carathéodory spaces seen from within
TL;DR: In this article, a local condition on curves is defined for a smooth manifold, where a subset H in the set of all piecewise smooth curves c in V is distinguished by local condition.
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A Cortical Based Model of Perceptual Completion in the Roto-Translation Space
Giovanna Citti,Alessandro Sarti +1 more
TL;DR: A mathematical model of perceptual completion and formation of subjective surfaces, which is at the same time inspired by the architecture of the visual cortex, and is the lifting in the 3-dimensional rototranslation group of the phenomenological variational models based on elastica functional is presented.
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A Comprehensive Introduction to Sub-Riemannian Geometry
TL;DR: Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints as discussed by the authors, where one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other.
Journal ArticleDOI
Optimal control in laser-induced population transfer for two- and three-level quantum systems
TL;DR: In this paper, the authors apply the techniques of control theory and of sub-Riemannian geometry to laser-induced population transfer in two-and three-level quantum systems, where the aim is to induce complete population transfer by one or two laser pulses minimizing the pulse fluences.
References
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A comprehensive introduction to differential geometry
TL;DR: Spivak's comprehensive introduction to differential geometry as discussed by the authors takes as its theme the classical roots of contemporary differential geometry, and explains why it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely to rigorize the concepts of classical differential geometry.
Journal ArticleDOI
Hypoelliptic second order differential equations
TL;DR: In this paper, it was shown that the sufficient conditions for hypoelliptie are not satisfied by the equation, and that the necessary conditions are not sufficient for all differential equations with constant coefficients.
Journal ArticleDOI
Hypoelliptic differential operators and nilpotent groups
TL;DR: In this paper, the authors present sufficient condi t ions for hypoe l l l ip t i c i ty... this paper..,.. The authors present a free Lie-A-Lgebras analysis on the p roof of the T h eo rem.
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Non-Commutative Geometry
TL;DR: For purely mathematical reasons, it is necessary to consider spaces which cannot be represented as point set sand where the coordinates describing the space do not commute as mentioned in this paper, i.e., spaces which are described by algebras of coordinates which are not commutative.