Search results for "22f30"

showing 3 items of 3 documents

Isometries of nilpotent metric groups

2016

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups and, in particular, Carnot groups equipped with Carnot-Carath\'eodory distances. We study the regularity of isometries, i.e., distance-preserving homeomorphisms. Our first result is the analyticity of such maps between metric Lie groups. The second result is that if two metric Lie groups are connected and nilpotent then every isometry between the groups is the composition of a left translation and an isomorphism.…

Mathematics - Differential GeometryIsometriesPure mathematicsA ne transformationsGeneral Mathematics22E25 53C30 22F30Group Theory (math.GR)01 natural sciencesisometriesMathematics - Metric GeometryetäisyysFOS: MathematicsMathematics (all)Mathematics::Metric GeometryA ne transformations; Isometries; Nilpotent groups; Nilradical; Mathematics (all)0101 mathematicsdistanceMathematicsLie groupsmatematiikkamathematicsta111010102 general mathematicsLie groupMetric Geometry (math.MG)nilpotent groupsnilradicalComposition (combinatorics)Manifoldaffine transformationsNilpotentDifferential Geometry (math.DG)Nilpotent groupsMetric (mathematics)IsometryNilradicalIsomorphismMathematics - Group TheoryCounterexampleJournal de l’École polytechnique — Mathématiques
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A Cornucopia of Carnot groups in Low Dimensions

2022

Abstract Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invaria…

Mathematics - Differential GeometryApplied Mathematicsnilpotent Lie algebrasLien ryhmätfree nilpotent groupsharmoninen analyysistratified groupsdifferentiaaligeometria510 MathematicsDifferential Geometry (math.DG)Carnot groupsFOS: Mathematicsexponential coordinatesGeometry and Topologyassociated Carnot-graded Lie algebra53C17 43A80 22E25 22F30 14M17Analysis
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A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

2017

AbstractCarnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks.We consider them as special cases of graded groups and as homogeneous metric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups.

Pure mathematicsmetric groupssub-finsler geometryengineering.material01 natural sciencesdifferentiaaligeometriasymbols.namesakesub-Finsler geometryMathematics::Metric Geometry0101 mathematics22f3014m17MathematicsPrimer (paint)QA299.6-433homogeneous groupshomogeneous spacesApplied Mathematics010102 general mathematics05 social sciencesryhmäteorianilpotent groupsCarnot groups; homogeneous groups; homogeneous spaces; metric groups; nilpotent groups; sub-Finsler geometry; sub-Riemannian geometry; Analysis; Geometry and Topology; Applied Mathematicssub-riemannian geometrysub-Riemannian geometry43a8053c17Carnot groupscarnot groupsengineeringsymbols22e25Geometry and Topology0509 other social sciences050904 information & library sciencesCarnot cycleAnalysisAnalysis and Geometry in Metric Spaces
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