Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study hypersurfaces with constant r  th mean curvature SrSr. We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an r  th Jacobi field of a hypersurface with Sr+1Sr+1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.