Yampolskiy O. Geometry of submanifolds in fiber bundles

This research focuses on submanifolds in fiber bundles with Sasaki metric mostly on minimal and totally geodesic. We have found the boundaries of the sectional curvature of the Sasaki metric of the sphere bundle over the space constant curvature. We have found the projections of the geodesic lines of the tangent (sphere) bundle with the Sasaki metric of classical space. We have found the conditions on totally geodesic property of lifts into the tangent bundle by a vector field along the submanifold of arbitrary codimension in the base. We have proved that the submanifold generated by characteristic vector field of the Sasakian structure is totally geodesic, particularly by the Hopf unit vector field on spheres. We have proved stability the Hopf unit vector field with respect to classical normal variations. We have found all totally geodesic left-invariant unit vector fields on three-dimensional Lie groups with left-invariant metric; the analysis of stability with respect to classical and some special variations is given. We have found the generalization of the Sasaki metric to the case of metriziable vector bundles with metric connection and proved the analogs of corresponding results for the unit tangent bundle.