Abstract
Given a system of vector fields on a smooth manifold that spans a plane field of constant rank, we present a systematic method and an algorithm to find submanifolds that are invariant under the flows of the vector fields. We present examples of partition into invariant submanifolds, which further gives partition into orbits. We use the method of generalized Frobenius theorem by means of exterior differential systems.
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References
Agrachev A, Sachkov Y. Control Theory from the Geometric Viewpoint. Berlin: Springer-Verlag, 2004
Ahn H, Han C K. Local geometry of Levi-forms associated with the existence of complex submanifolds and the minimality of generic CR manifolds. J Geom Anal, 2012, 22: 561–582
Brockett R, Millman R, Sussmann H. Differential Geometric Control Theory. Boston: Birkhäuser, 1983
Bryant R, Chern S-S, Gardner R, et al. Exterior Differential Systems. New York: Springer-Verlag, 1991
Chow WL. Über systeme von linearen partiellen differentialgleichungen erster ordnung. Math Ann, 1939, 117: 98–105
Gardner R. Invariants of Pfaffian systems. Trans Amer Math Soc, 1967, 126: 514–533
Han C K. Pfaffian systems of Frobenius type and solvability of generic overdetermined PDE systems. In: Symmetries and Overdetermined Systems of Partial Differential Equations, vol. 144. New York: Springer, 2008, 421–429
Han C K. Generalization of the Frobenius theorem on involutivity. J Korean Math Soc, 2009, 46: 1087–1103
Han C K. Foliations associated with Pfaffian systems. Bull Korean Math Soc, 2009, 46: 931–940
Han C K, Kim H. Holomorphic functions on almost complex manifolds. J Korean Math Soc, 2012, 49: 379–394
Han C K, Lee K H. Integrable submanifolds in almost complex manifolds. J Geom Anal, 2010, 20: 177–192
Han C K, Park J D. Partial integrability of almost complex structures and the local solvability of quasi-linear Cauchy- Riemann equations. Pacific J Math, 2013, 265: 59–84
Han C K, Park J D. Method of characteristics and first integrals for systems of quasi-linear partial differential equations of first order. Sci China Math, 2015, 58: 1665–1676
Han C K, Park J D. Quasi-linear partial differential equations of first order with Cauchy data of higher codimensions. J Math Anal Appl, 2015, 430: 390–402
Ivey T A, Landsberg J M. Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems. Providence, RI: Amer Math Soc, 2003
Jurdjevic V. Geometric Control Theory. Cambridge: Cambridge University Press, 1997
Krener A. A generalization of Chow’s theorem and the bang-bang theorem to non-linear control problems. SIAM J Control, 1974, 12: 43–52
Milnor J, Stasheff D. Characteristic Classes. Princeton: Princeton University Press, 1974
Sussmann H. Orbits of families of vector fields and integrability of distributions. Trans Amer Math Soc, 1973, 180: 171–188
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Ahn, H., Han, C. Invariant submanifolds for systems of vector fields of constant rank. Sci. China Math. 59, 1417–1426 (2016). https://doi.org/10.1007/s11425-016-5139-0
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DOI: https://doi.org/10.1007/s11425-016-5139-0