6533b7dafe1ef96bd126ecb3

RESEARCH PRODUCT

Isometries of nilpotent metric groups

Ville KiviojaEnrico Le Donne

subject

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 TheoryCounterexample

description

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. There are counterexamples if one does not assume the groups to be either connected or nilpotent. The first result is based on a solution of the Hilbert 5th problem by Montgomery and Zippin. The second result is proved, via the first result, considering the Riemannian case, which for self-isometries was solved by Wolf.

yearjournalcountryeditionlanguage
2016-01-29Journal de l’École polytechnique — Mathématiques
https://doi.org/10.5802/jep.48