  We study Riemannian manifolds with boundary under a lower Ricci curvature
bound, and a lower mean curvature bound for the boundary. We prove a volume
comparison theorem of Bishop-Gromov type concerning the volumes of the metric
neighborhoods of the boundaries. We conclude several rigidity theorems. As one
of them, we obtain a volume growth rigidity theorem. We also show a splitting
theorem of Cheeger-Gromoll type under the assumption of the existence of a
single ray.
