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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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.
- [1] Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
- [2] G. A. Bliss, Lectures on the calculus of variations, Phoenix Science Series, 1963.
- [3] Élie Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques, Actualités Sci. Ind., no. 994, Hermann et Cie., Paris, 1945 (French). MR 0016174
- [4] S. S. Chern, Minimal submanifolds in a Riemannian manifold, University of Kansas, Department of Mathematics Technical Report 19 (New Series), Univ. of Kansas, Lawrence, Kan., 1968. MR 0248648
- [5] Robert Hermann, E. Cartan’s geometric theory of partial differential equations, Advances in Math. 1 (1965), no. fasc. 3, 265–317. MR 0209623, https://doi.org/10.1016/0001-8708(65)90040-X
- [6] Robert Hermann, The second variation of minimal submanifolds, J. Math. Mech. 16 (1966), 473–491. MR 0208538
- [7] E. Kähler, Einführung in die Theorie der System von Differentialgleichungen, Chelsea, New York, 1949.
- [8] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR 0152974
- [9] M. Kuranishi, Lectures on exterior differential systems, Tata Institute of Fundamental Research, Bombay, 1962.
- [10] Alan B. Poritz, A generalization of parallelism in Riemannian geometry, the 𝐶^{𝜔} case, Trans. Amer. Math. Soc. 152 (1970), 461–494. MR 0268813, https://doi.org/10.1090/S0002-9947-1970-0268813-2
- [11] Yozô Matsushima, On a theorem concerning the prolongation of a differential system, Nagoya Math. J. 6 (1953), 1–16. MR 0058817
- [12] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 0233295, https://doi.org/10.2307/1970556
Retrieve articles in Transactions of the American Mathematical Society with MSC: 58A15
Retrieve articles in all journals with MSC: 58A15
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