Symmetries of Flat Rank Two Distributions and Sub-riemannian Structures

Flat sub-Riemannian structures are local approximations — nilpotentizations — of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5. 1. Sub-Riemannian structures A sub-Riemannian geometry is a triple (M,∆, 〈·, ·〉), where M is a smooth manifold, ∆ ⊂ TM is a smooth distribution on M , ∆ = {∆q ⊂ TqM | q ∈ M}, and 〈·, ·〉 is an inner product in ∆ that smoothly depends on a point in M , 〈·, ·〉 = {〈·, ·〉q— an inner product in ∆q | q ∈M}. The pair (∆, 〈·, ·〉) is a sub-Riemannian structure on M ; if dimM = n and dim ∆q = k, q ∈ M , then we say that (∆, 〈·, ·〉) is a (k, n)-structure. The number k is called the rank of the distribution ∆ or the structure (∆, 〈·, ·〉). In this work, we are interested in the special class of distributions and sub-Riemannian structures called flat. Let G be a connected simply connected nilpotent Lie group. Suppose that its Lie algebra g is graded: g = g ⊕ g ⊕ · · · ⊕ g, [g, g ] ⊂ g , g = {0} ∀r > s, and generated as a Lie algebra by its component of degree 1: Lie(g) = g. Then ∆ = g can be considered as a completely nonintegrable (bracket-generating) left-invariant distribution on the Lie group G. We call such a distribution ∆ flat. Further, if ∆ is equipped with a left-invariant inner product 〈·, ·〉 obtained from an inner product in g, then (∆, 〈·, ·〉) is called a flat sub-Riemannian structure on G. Flat sub-Riemannian structures arise as local approximations — nilpotentizations — of arbitrary sub-Riemannian structures at regular points (see [2], [3] for details). Received by the editors May 4, 2001. 2000 Mathematics Subject Classification. Primary 53C17.