Isometries Groups and a Multiresolution Analysis on Sub-Riemannian Manifolds

TL;DR

This paper explores the relationship between isometry groups of Riemannian and sub-Riemannian manifolds, establishing fixed point results and constructing multiresolution analyses to develop Haar bases on these manifolds.

Contribution

It introduces a novel connection between isometries of Riemannian and sub-Riemannian structures, enabling the creation of Haar bases through multiresolution analysis on sub-Riemannian manifolds.

Findings

Established fixed point results for isometry groups.

Defined a multiresolution analysis on sub-Riemannian manifolds.

Constructed Haar bases on these manifolds.

Abstract

In this letter we exhibit the relation between the isometries of a Riemannian contraction of a sub-Riemannian manifold and those of the sub-Riemannian metric, for to use this relation with two goals: establishing a result about the existence of fixed points of isometries groups; and the other, defining a Multiresolution Analysis (MRA) on sub-Riemannian manifolds that it will permit to obtain Haar's bases on the manifolds before mentioned. Keywords: Sub-Riemannian geometry, minimizing geodesic, Haar functions, self-similarity.