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

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

2007-12-13

arxiv.org/abs/0712.2132v1

Mathematics

Differential Geometry

19 pages, 2 figures

Scientific paper

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.

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