  We study the Jacobi osculating rank of geodesics on naturally reductive
homogeneous manifolds and we apply this theory to the 3-dimensional case. Here,
each non-symmetric, simply connected naturally reductive 3-manifold can be
given as a principal bundle over a surface of constant curvature, such that the
curvature of its horizontal distribution is also a constant. Then, we prove
that the Jacobi osculating rank of every geodesic is two except for the Hopf
fibers, where it is zero. Moreover, we determine all isotropic geodesics and
the isotropic tangent conjugate locus.
