  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.
