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

Abstract

AbstractSuppose 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 $$q\in M$$ q ∈ M denote by $$\mathfrak {i}^*(q)$$ i ∗ ( q ) the Lie algebra of germs at q of infinitesimal isometries of (M, H, g). It is proved that for a generic point $$q\in M$$ q ∈ M , $$\dim \mathfrak {i}^*(q)$$ dim i ∗ ( q ) can only assume the values 1, 2, 4. Moreover, $$\dim \mathfrak {i}^*(q) = 4$$ dim 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.