Extremals in some classes of Carnot groups

Let G be a Carnot group and D = {e ₁, e ₂} be a bracket generating left invariant distribution on G. In this paper, we obtain two main results. We first prove that there only exist normal minimizers in G if the type of D is (2, 1, ??, 1) or (2, 1, ??, 1, 2). This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds. As one corollary, we also obtain that there are only normal minimizers when dimG ? 5. We construct a class of Carnot groups such as that of type (2, 1, ??, 1, 2, n ₀, ??, n _a ) with n ₀ ? 1, n _i ? 0, i = 1, ??, a, in which there exist strictly abnormal extremals. This implies that, for any given manifold of dimension n ? 6, we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers. We conclude that the dimension n = 5 is the border line for the existence and nonexistence of strictly abnormal extremals. Our main technique is based on the equations for the normal and abnormal extremals.