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

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