Deformations of integrals of exterior differential systems

Author:
Dominic S. P. Leung

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

MSC:
Primary 58A15

MathSciNet review:
0314082

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Abstract: On any general solution of an exterior differential system , a system of linear differential equations, called the equations of variation of , is defined. Let be a vector field defined on a general solution of such that it satisfies the equations of variation and wherever it is defined, 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, is locally the deformation vector field of a one-parameter family of general solutions of . 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

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

Keywords:
Exterior differential system,
Cartan-Kähler theory,
general solution,
regular integral chain,
one-parameter family of integral manifolds,
deformation vector,
equations of variation,
-field,
normal system,
normal solution,
normal data,
associated differential systems,
minimal submanifolds,
Jacobi field

Article copyright:
© Copyright 1972
American Mathematical Society