Deformations of integrals of exterior differential systems

Author:
Dominic S. P. Leung

Journal:
Trans. Amer. Math. Soc. **170** (1972), 333-358

MSC:
Primary 58A15

DOI:
https://doi.org/10.1090/S0002-9947-1972-0314082-6

MathSciNet review:
0314082

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Abstract | References | Similar Articles | Additional Information

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-KĂ€hler 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.

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Additional Information

Keywords:
Exterior differential system,
Cartan-Kähler theory,
general solution,
regular integral chain,
one-parameter family of integral manifolds,
deformation vector,
equations of variation,
<IMG WIDTH="16" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$I$">-field,
normal system,
normal solution,
normal data,
associated differential systems,
minimal submanifolds,
Jacobi field

Article copyright:
© Copyright 1972
American Mathematical Society