Communications in Analysis and Geometry

Volume 29 (2021)

Number 1

Model spaces in sub-Riemannian geometry

Pages: 77 – 113

DOI: https://dx.doi.org/10.4310/CAG.2021.v29.n1.a3

Author

Erlend Grong (Université Paris Sud, Laboratoire des Signaux et Systèmes (L2S), Supélec CNRS, Université Paris-Saclay, Gif-sur-Yvette, France; and Department of Mathematics, University of Bergen, Norway)

Abstract

We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry. We will show that these spaces have a canonical choice of partial connection on their horizontal bundle, which is determined by isometries and generalizes the Levi–Civita connection for the special case of Riemannian model spaces. The number of invariants needed to describe model spaces with the same tangent cone is in general greater than one, and these invariants are not necessarily related to the holonomy of the canonical connections.

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Received 8 November 2017

Accepted 6 April 2018

Published 11 March 2021