Search results for "homogeneous groups"

showing 3 items of 3 documents

Radon–Nikodym Property and Area Formula for Banach Homogeneous Group Targets

2013

We prove a Rademacher-type theorem for Lipschitz mappings from a subset of a Carnot group to a Banach homogeneous group, equipped with a suitably weakened Radon-Nikodym property. We provide a metric area formula that applies to these mappings and more generally to all almost everywhere metrically differentiable Lipschitz mappings defined on a Carnot group. peerReviewed

Discrete mathematicsMathematics::Functional AnalysisProperty (philosophy)General Mathematicsmetric area formulata111Mathematics::Analysis of PDEsCarnot groupBanach homogeneous groupsalmost everywhere differentiabilityRadon-Nikodym propertyLipschitz continuityRadon–Nikodym theoremBanach homogeneous groups; metric area formula; almost everywhere differentiability; Radon-Nikodym propertyMetric (mathematics)Homogeneous groupMathematics::Metric GeometryAlmost everywhereDifferentiable functionMathematics
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Regularity properties of spheres in homogeneous groups

2015

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an alg…

Mathematics - Differential GeometryPure mathematicsGeodesicjoukot (matematiikka)General MathematicsGroup Theory (math.GR)algebra01 natural sciencessets (mathematics)Homothetic transformationMathematics - Metric Geometry0103 physical sciencesEuclidean geometryFOS: MathematicsHeisenberg groupMathematics::Metric GeometryMathematics (all)spheres0101 mathematicsMathematics28A75 22E25 53C60 53C17 26A16homogeneous groupsmatematiikkamathematicsGroup (mathematics)Applied Mathematicsta111010102 general mathematicsLie groupMetric Geometry (math.MG)Lipschitz continuityAutomorphismDifferential Geometry (math.DG)regularity properties010307 mathematical physicsMathematics - Group TheoryMathematics (all); Applied Mathematics
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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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