LAUSR

We prove that there do not exist quasi-isometric embeddings of connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics into a metric measure space satisfying the curvature-dimension condition RCD (0, N ) with N ∈ ℝ and N > 1. In fact, we can prove that a sub-Riemannian manifold whose generic degree of nonholonomy is not smaller than 2 cannot be bi-Lipschitzly embedded in any Banach space with the Radon-Nikodym property. We also get that every regular sub-Riemannian manifold do not satisfy the curvature-dimension condition CD ( K, N ), where K, N ∈ ℝ and N > 1. Along the way to the proofs, we show that the minimal weak upper gradient and the horizontal gradient coincide on the Carnot-Carath´eodory spaces which may have independent interests.