A topological splitting theorem for sub-Riemannian manifolds

We prove an analogue of the Cheeger-Gromoll splitting theorem for sub-Riemannian manifolds with the measure contraction property instead of the nonnegativity of the Ricci curvature. If such a sub-Riemannian manifold contains a straight line, then the manifold splits diffeomorphically, where the splitting is not necessarily isometric. We prove that such a sub-Riemannian manifold containing a straight line cannot split isometrically under some typical condition in sub-Riemannian geometry. Heisenberg groups are such examples.