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http://dx.doi.org/10.4134/JKMS.2009.46.5.1087

GENERALIZATION OF THE FROBENIUS THEOREM ON INVOLUTIVITY  

Han, Chong-Kyu (DEPARTMENT OF MATHEMATICS SEOUL NATIONAL UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.46, no.5, 2009 , pp. 1087-1103 More about this Journal
Abstract
Given a system of s independent 1-forms on a smooth manifold M of dimension m, we study the existence of integral manifolds by means of various generalized versions of the Frobenius theorem. In particular, we present necessary and sufficient conditions for there to exist s'-parameter (s' < s) family of integral manifolds of dimension p := m-s, and a necessary and sufficient condition for there to exist integral manifolds of dimension p', p' $\leq$ p. We also present examples and applications to complex analysis in several variables.
Keywords
Pfaffian system; involutivity; integral manifold; foliation;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
Times Cited By Web Of Science : 3  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
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1 R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, Exterior Differential Systems, Springer-Verlag, New York, 1986
2 R. Bryant, Exterior differential system, Lectures at Duke Univ. noted by Sungho Wang
3 E. Cartan, Lecons sur les invariants $int{\acute{e}}graux$, Hermann, Paris, 1922
4 J. S. Cho and C. K. Han, Complete prolongation and the Frobenius integrability for overdetermined systems of partial differential equations, J. Korean Math. Soc. 39 (2002), no. 2, 237–252   과학기술학회마을   DOI
5 E. Cartan, Les $syst{\grave{e}}mes$ $diff{\acute{e}}rentiels$ $ext{\acute{e}}rieurs$ et leurs applications, $g{\acute{e}}om{\acute{e}}triques$ Hermann, 1971 Photocopy Paris, 1945
6 F. Deahna, ${\ddot{U}}ber$ die Bedingungen der Integrabilitat, J. Reine und Angew. Math. 20 (1840), 340–350
7 C. K. Han, Solvability of overdetermined PDE systems that admit a complete prolongation and some local problems in CR geometry, J. Korean Math. Soc. 40 (2003), no. 4, 695–708   과학기술학회마을   DOI   ScienceOn
8 S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271   DOI
9 A. Clebsch, ${\ddot{U}}ber$ die simultane Integration linearer partieller Differentialgleichungen, J. Reine und Angew. Math. (Crelle) 65 (1866), 257–268
10 G. Darboux, Sur le $probl{\grave{e}}me$ de Pfaff (1), (2), Bull. Sci. Math. 6 (1882), 14–36, 49–68
11 G. Frobenius, ${\ddot{U}}ber$ das Pfaffsche probleme, J. Reine und Angew. Math. 82 (1877), 230–315
12 P. Griffiths and G. Jensen, Differential Systems and Isometric Embeddings, Annals of Mathematics Studies, 114. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1987