International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 9, Pages 541-557

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

Ruslan A. Sharipov

Rabochaya Street 5, Ufa 450003, Russia

Received 21 January 2001; Revised 26 June 2001

Copyright © 2002 Ruslan A. Sharipov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 along geodesic lines. Normality of shift means that moving hypersurface keeps 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 force F acting 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 field F of dynamical system should satisfy some system of partial differential equations. Recently, this system of equations was integrated, and explicit formula for F was obtained. But this formula is local. The main goal of this paper is to reveal global geometric structures associated with local expressions for F given by explicit formula.