## Deformations of integrals of exterior differential systems

HTML articles powered by AMS MathViewer

- 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.## References

- Richard L. Bishop and Richard J. Crittenden,
*Geometry of manifolds*, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR**0169148**
G. A. Bliss, - Ălie Cartan,
*Les systĂšmes diffĂ©rentiels extĂ©rieurs et leurs applications gĂ©omĂ©triques*, ActualitĂ©s Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 994, Hermann & Cie, Paris, 1945 (French). MR**0016174** - S. S. Chern,
*Minimal submanifolds in a Riemannian manifold*, University of Kansas, Department of Mathematics Technical Report 19 (New Series), University of Kansas, Lawrence, Kan., 1968. MR**0248648** - Robert Hermann,
*E. Cartanâs geometric theory of partial differential equations*, Advances in Math.**1**(1965), no.Â fasc. 3, 265â317. MR**209623**, DOI 10.1016/0001-8708(65)90040-X - Robert Hermann,
*The second variation of minimal submanifolds*, J. Math. Mech.**16**(1966), 473â491. MR**0208538**, DOI 10.1512/iumj.1967.16.16032
E. KĂ€hler, - Shoshichi Kobayashi and Katsumi Nomizu,
*Foundations of differential geometry. Vol I*, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR**0152974**
M. Kuranishi, - Alan B. Poritz,
*A generalization of parallelism in Riemannian geometry, the $C^{\omega }$ case*, Trans. Amer. Math. Soc.**152**(1970), 461â494. MR**268813**, DOI 10.1090/S0002-9947-1970-0268813-2 - YozĂŽ Matsushima,
*On a theorem concerning the prolongation of a differential system*, Nagoya Math. J.**6**(1953), 1â16. MR**58817** - James Simons,
*Minimal varieties in riemannian manifolds*, Ann. of Math. (2)**88**(1968), 62â105. MR**233295**, DOI 10.2307/1970556

*Lectures on the calculus of variations*, Phoenix Science Series, 1963.

*EinfĂŒhrung in die Theorie der System von Differentialgleichungen*, Chelsea, New York, 1949.

*Lectures on exterior differential systems*, Tata Institute of Fundamental Research, Bombay, 1962.

## 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