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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

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A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

Enrico Le Donne
Published Online: 2018-01-04 | DOI: https://doi.org/10.1515/agms-2017-0007

Abstract

Carnot 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.

Keywords : Carnot groups; sub-Riemannian geometry; sub-Finsler geometry; homogeneous spaces; homogeneous groups; nilpotent groups; metric groups

MSC 2010: 53C17; 43A80; 22E25; 22F30; 14M17

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About the article

Received: 2016-11-18

Revised: 2017-09-07

Accepted: 2017-11-08

Published Online: 2018-01-04


Citation Information: Analysis and Geometry in Metric Spaces, Volume 5, Issue 1, Pages 116–137, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2017-0007.

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