Application of codimension one foliation in Zermelo's problem on Riemannian manifolds

We research the Zermelo navigation problem on Riemannian manifolds in dim⁡(R×M)=3dim⁡(R×M)=3 under the force representing the action of the perturbing “wind” distribution modeled by the vector field on manifold M  . We consider a fibered manifold π:R×M→Rπ:R×M→R where M   represents the “sea”, RR represents the time and π is the first canonical projection. We perturb it by a time-(in)dependent vector field on the manifold M with a codimension one foliation. We aim to research the behavior of the paths in reference to their optimization. Geometrically we find the deviation of geodesics under the action of a time-dependent vector field. We present corresponding simulation with the whirlpool perturbation. We also consider a new concept of real application in the navigational decision support systems.