  In this paper, we consider critical maps of a horizontal energy functional
for maps from a sub-Riemannian manifold to a Riemannian manifold. These
critical maps are referred to as subelliptic harmonic maps. In terms of the
subelliptic harmonic map heat flow, we investigate the existence problem for
subelliptic harmonic maps. Under the assumption that the target Riemannian
manifold has non-positive sectional curvature, we prove some Eells-Sampson type
existence results for this flow when the source manifold is either a step-2
sub-Riemannian manifold or a step-r sub-Riemannian manifold whose
sub-Riemannian structure comes from a tense Riemannian foliation. Finally, some
Hartman type results are also established for the flow.
