  We show that, for a closed orientable n-manifold, with n not congruent to 3
modulo 4, the existence of a CR-regular embedding into complex (n-1)-space
ensures the existence of a totally real embedding into complex n-space. This
implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire
semi-characteristic possesses no CR-regular embedding into complex 4k-space. We
also pay special attention to the cases of CR-regular embeddings of spheres and
of simply-connected 5-manifolds.
