Deformations of integrals of exterior differential systems

Abstract:

On any general solution of an exterior differential system $I$, a system of linear differential equations, called the equations of variation of $I$, is defined. Let ${\text {v}}$ be a vector field defined on a general solution of $I$ such that it satisfies the equations of variation and wherever it is defined, ${\text {v}}$ is either the zero vector or it is not tangential to the general solution. By means of some associated differential systems and the fundamental theorem of Cartan-Kähler theory, it is proved that, under the assumption of real analyticity, ${\text {v}}$ is locally the deformation vector field of a one-parameter family of general solutions of $I$. As an application, it is proved that, under the assumption of real analyticity, every Jacobi field on a minimal submanifold of a Riemannian manifold is locally the deformation vector field of a one-parameter family of minimal submanifolds.