On the dimension of the algebras of local infinitesimal isometries of 3-dimensional special sub-Riemannian manifolds

Abstract

Suppose that we are given a contact sub-Riemannian manifold (M, H, g) of dimension 3 such that the Reeb vector field is an infinitesimal isometry (such manifolds will be referred to as special). For a point qin M denote by mathfrak {i}^*(q) the Lie algebra of germs at q of infinitesimal isometries of (M, H, g). It is proved that for a generic point qin M, dim mathfrak {i}^*(q) can only assume the values 1, 2, 4. Moreover, dim mathfrak {i}^*(q) = 4 if and only if the curvature function determined by the canonical sub-Riemannian connection is constant in a neighborhood of q. The latter case is possible if (M, H, g) is locally isometric, in a neighborhood of q, to a left-invariant sub-Riemannian structure on a 3-dimensional Lie group.