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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $ 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.


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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, $ I$-field, normal system, normal solution, normal data, associated differential systems, minimal submanifolds, Jacobi field
Article copyright: © Copyright 1972 American Mathematical Society