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Transactions of the American Mathematical Society

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Orbits of families of vector fields and integrability of distributions


Author: Héctor J. Sussmann
Journal: Trans. Amer. Math. Soc. 180 (1973), 171-188
MSC: Primary 58A30; Secondary 53C10
DOI: https://doi.org/10.1090/S0002-9947-1973-0321133-2
MathSciNet review: 0321133
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Abstract: Let $ D$ be an arbitrary set of $ {C^\infty }$ vector fields on the $ {C^\infty }$ manifold $ M$. It is shown that the orbits of $ D$ are $ {C^\infty }$ submanifolds of $ M$, and that, moreover, they are the maximal integral submanifolds of a certain $ {C^\infty }$ distribution $ {P_D}$. (In general, the dimension of $ {P_D}(m)$ will not be the same for all $ m \in M$.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow's theorem to the maximal integral submanifolds of the smallest distribution $ \Delta $ such that every vector field $ X$ in the Lie algebra generated by $ D$ belongs to $ \Delta $ (i.e. $ X(m) \in \Delta (m)$ for every $ m \in M$). Their work therefore requires the additional assumption that $ \Delta $ be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of $ \Delta $ is not assumed in proving the first main result. It turns out that $ \Delta $ is integrable if and only if $ \Delta = {P_D}$, and this fact makes it possible to derive a characterization of integrability and Chow's theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.


References [Enhancements On Off] (What's this?)

  • [1] C. Chevalley, Theory of Lie groups. I, Princeton Math. Ser., vol. 8, Princeton Univ. Press, Princeton, N. J., 1946. MR 7, 412. MR 0082628 (18:583c)
  • [2] W. L. Chow, Über Systeme von linearen partiellen Differential-gleichungen erster Ordnung, Math. Ann. 117 (1939), 98-105. MR 1, 313. MR 0001880 (1:313d)
  • [3] S. Helgason, Differential geometry and symmetric spacer, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962, MR 26 #2986. MR 0145455 (26:2986)
  • [4] R. Hermann, On the accessibility problem in control theory, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 325-332. MR 26 #6891. MR 0149402 (26:6891)
  • [5] C. Lobry, Contrôlabilité des systèmes non linéaires, SIAM J. Control 8 (1970), 573-605. MR 42 #6860. MR 0271979 (42:6860)
  • [6] M. Matsuda, An integration theorem for completely integrable systems with singularities, Osaka J. Math. 5 (1968), 279-283. MR 39 #4876. MR 0243555 (39:4876)
  • [7] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398-404. MR 33 #8005. MR 0199865 (33:8005)
  • [8] H. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations 12 (1972), 95-116. MR 0338882 (49:3646)

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DOI: https://doi.org/10.1090/S0002-9947-1973-0321133-2
Article copyright: © Copyright 1973 American Mathematical Society

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