Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds

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 M3(κ,τ) over a surface of constant curvature κ, such that the curvature of its horizontal distribution is a constant τ>0, with τ2≠κ. Then, we prove that the Jacobi osculating rank of every geodesic of M3(κ,τ) is two except for the Hopf fibers, where it is zero. Moreover, we determine all isotropic geodesics and the isotropic tangent conjugate locus.