Global geometric structures associated with dynamical systems admitting normal shift of hypersurfaces in Riemannian manifolds

摘要

One of the ways of transforming hypersurfaces in Riemannian manifold is to move their points along some lines. In Bonnet construction of geodesic normal shift, these points move alonggeodesic lines. Normality of shift means that moving hypersurfacekeeps orthogonality to the trajectories of all its points. Geodesic lines correspond to the motion of free particles if the points of hypersurface are treated as physical entities obeying Newton's second law. An attempt to introduce some external forceFacting on the points of moving hypersurface in Bonnet construction leads to the theory of dynamical systems admitting a normal shift. As appears in this theory, the force fieldFof dynamical system should satisfy some system of partial differential equations. Recently, this system of equations was integrated, and explicit formula forFwas obtained. But this formula is local. The main goal of this paper is to reveal global geometric structures associated with local expressions forFgiven by explicit formula.