Heat flow for horizontal harmonic maps into a class of Carnot-Caratheodory spaces

Let X and B be two Riemannian manifolds with π : X → B being a Riemannian submersion. Let H be the corresponding horizontal distribution, which is perpendicular to the tangent bundle of the fibres of π : X → B. Then X (just considered as a differentiable manifold), together with the distribution H, forms a so-called Carnot-Caratheodory space [1], when the Riemannian metric of X is restricted to H. On X, as a Carnot-Caratheodory space, can then be defined the notions of Carnot-Caratheodory distance (sometimes called sub-Riemannian distance), (minimizing) geodesic, completeness (under the Carnot-Caratheodory distance), etc; a geodesic is actually a horizontal curve which locally realizes the Carnot-Caratheodory distance. In this note, we always assume that X is complete, as both a Riemannian manifold and a Carnot-Caratheodory space, and the Riemannian submersion π : X → B together with its horizontal distribution H satisfies the following conditions 1) the Chow condition: the vector fields ofH X1, X2, · · · , and their iterated Lie brackets [Xi, Xj], [[Xi, Xj], Xk], · · · span the tangent space TxX at every point of X; 2) the sectional curvature of X (as a Riemannian manifold) in the direction of H is non-positive.