11 - Grushin Manifolds

Summary

Definition and Examples

A Grushin manifold is roughly speaking a Riemannian manifold endowed with a singular Riemannian metric. In this case the horizontal distribution does not play an important role, since the distribution coincides with the tangent bundle on the set of regular points. The horizontal curves do not make much sense either in this case. A Grushin manifold might be considered as a sub-Riemannian manifold where the distribution has the rank equal to the dimension of the space. We study this type of manifold here since it behaves similarly with some of the examples studied in the previous chapters. They are also closely related with a certain type of subelliptic operators, called Grushin operators.

Definition 11.1.1.Let M be a manifold of dimension n and let X₁,…, X_n be n vectors on M. Let S ={p ∈ M; span{X₁,…, X_n} = T_p M}. A point p ∈ S is called singular, while a point p ∉ S is called regular. Consider the Riemannianc metric g defined on the set of regular points M\S such that g(X_i, X_j) = δ_ij. Then (M, X_i, g) is called a Grushin manifold.

Recall that the step at a point p ∈ M is equal to 1 plus the number of Lie brackets of vector fields X_i needed to span the tangent space T_pM.