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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Deformations of integrals of exterior differential systems
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by Dominic S. P. Leung PDF
Trans. Amer. Math. Soc. 170 (1972), 333-358 Request permission

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
  • © Copyright 1972 American Mathematical Society
  • 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