Foliations on closed geodesics with positive mixed sectional curvature and special behaviour of $L$-Jacobi fields

The Hopf fibrations on spheres and real projective spaces with standard metrics serve the examples of odd-dimensional Riemannian manifolds with foliations on closed geodesies, that have positive constant sectional curvature in mixed directions. One can show, using the known equation of Jacobi vector fields in case of constant sectional curvature, that even-dimensional sphere and real projective space do not admit foliations on closed geodesies even locally, i.e., in neighborhood of any such geodesic. The same holds for multi-dimensional ruled submariifolds in spheres and real projective spaces with one-dimensional ruling and positive mixed sectional curvature. However, for any $\delta\in(0,1)$ there exists even-dimensional Riemannian manifold with foliation on closed geodesies, that have positive $\delta$-pinched mixed sectional curvature. By above reasons we studied foliations with positive mixed curvature and some additional conditions.
The aim of the paper is to prove Theorem 1, that if a Riemannian manifold admits a foliation on closed geodesies with positive mixed sectional curvature, then under some additional conditions on behaviour of $L$-Jacobi vector fields (that hold in a real projective space) and turbulence, the dimension of a manifold is odd.