Search results for "14M17"

showing 5 items of 5 documents

Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces

2015

We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in `Fractured fractals and broken dreams' by David and Semmes, or equivalently, Question 22 and hence also Question 24 in `Thirty-three yes or no questions about mappings, measures, and metrics' by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors $4$-regular metric space $X$ having big pieces of itself such that no Lipschitz map from a subset of $X$ to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from the Heisenberg group to the space $X$ having as image the whole $X$. As part of proving the above re…

53C17 22F50 22E25 14M17General MathematicsSpace (mathematics)Heisenberg group01 natural sciencesMeasure (mathematics)Image (mathematics)Set (abstract data type)Ahlfors-regular distancesMathematics - Metric Geometry53C170103 physical sciencesClassical Analysis and ODEs (math.CA)FOS: MathematicsHeisenberg groupMathematics::Metric GeometryMathematics (all)22E250101 mathematicsMathematicsDiscrete mathematicsmatematiikkamathematicsMathematics::Complex Variables010308 nuclear & particles physicsta111010102 general mathematicsMetric Geometry (math.MG)Lipschitz continuityMetric spaceMathematics - Classical Analysis and ODEsBounded function14M17; 22E25; 22F50; 53C17; Mathematics (all)14M1722F50
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Real structures on nilpotent orbit closures

2021

We determine the equivariant real structures on nilpotent orbits and the normalizations of their closures for the adjoint action of a complex semisimple algebraic group on its Lie algebra.

Mathematics - Algebraic Geometryreal form14R20 14M17 14P99 11S25 20G20homogeneous spaceMathematics::Rings and Algebrasreal structureGalois cohomology[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]FOS: MathematicsNilpotent orbitMathematics::Representation TheoryAlgebraic Geometry (math.AG)
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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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Sard property for the endpoint map on some Carnot groups

2016

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizat…

Mathematics - Differential Geometry0209 industrial biotechnologyPure mathematics53C17 22F50 22E25 14M17SubvarietyGroup Theory (math.GR)02 engineering and technologySard's property01 natural sciencesSet (abstract data type)020901 industrial engineering & automationAbnormal curves; Carnot groups; Endpoint map; Polarized groups; Sard's property; Sub-Riemannian geometry; Analysis; Mathematical PhysicsMathematics - Metric GeometryFOS: MathematicsPoint (geometry)Canonical mapAbnormal curves; Carnot groups Endpoint map Polarized groups Sard's property Sub-Riemannian geometry Analysis0101 mathematicsMathematics - Optimization and ControlMathematical PhysicsMathematicsApplied Mathematics010102 general mathematicsta111Polarized groupsCarnot groupLie groupEndpoint mapMetric Geometry (math.MG)Base (topology)ManifoldSub-Riemannian geometryDifferential Geometry (math.DG)Optimization and Control (math.OC)Carnot groupsAbnormal curvesMathematics - Group TheoryAnalysis
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