Search results for " Carnot groups"

showing 2 items of 2 documents

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
researchProduct

Extremal polynomials in stratified groups

2018

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations.

Statistics and Probabilityextremal polynomialsMathematics - Differential GeometryPure mathematicsGeodesicStructure (category theory)Group Theory (math.GR)Characterization (mathematics)algebra01 natural sciencesdifferentiaaligeometriaMathematics - Analysis of PDEsMathematics - Metric Geometry53C17FOS: Mathematics0101 mathematicsAlgebraic numberMathematics - Differential Geometry; Mathematics - Differential Geometry; Mathematics - Analysis of PDEs; Mathematics - Group Theory; Mathematics - Metric Geometry; Mathematics - Optimization and Control; 53C17; 49K30; 17B70Mathematics - Optimization and ControlMathematics010102 general mathematicsStatisticsta111polynomitProlongation53C17 49K30 17B70Lie groupMetric Geometry (math.MG)abnormal extremals010101 applied mathematicsNilpotent Lie algebraNilpotentsub-Riemannian geometryabnormal extremals extremal polynomials Carnot groups sub-Riemannian geometryAbnormal extremals; Carnot groups; Extremal polynomials; Sub-Riemannian geometry; Analysis; Statistics and Probability; Geometry and Topology; Statistics Probability and UncertaintyDifferential Geometry (math.DG)Optimization and Control (math.OC)Carnot groups17B70Probability and UncertaintyGeometry and TopologyStatistics Probability and UncertaintyMathematics - Group TheoryAnalysisAnalysis of PDEs (math.AP)Mathematics - Differential Geometry; Mathematics - Differential Geometry; Mathematics - Analysis of PDEs; Mathematics - Group Theory; Mathematics - Metric Geometry; Mathematics - Optimization and Control; 53C17 49K30 17B7049K30
researchProduct