Abstract
In this paper, we study basic differential invariants of the pair (vector field, foliation). As a result, we establish a dynamic interpretation and a generalization of the Levi-Civita connection and Riemannian curvature treated as invariants of the geodesic flow on the tangent bundle.
References
A. A. Agrachev and R. V. Gamkrelidze, “Symplectic methods in optimization and control,” in: Geometry of Feedback and Optimal Control, Marcel Dekker (1998), pp. 19–77.
A. A. Agrachev and R. V. Gamkrelidze, “Feedback-invariant optimal control theory and diferential geometry-I. Regular extremals,” J. Dyn. Contr. Syst., 3, No. 3, 343–389 (1997).
A. A. Agrachev, “Feedback-invariant optimal control theory and diferential geometry-I. Jacobi curves for singular extremals,” J. Dyn. Contr. Syst., 4, 583–604 (1998).
A. A. Agrachev and I. Zelenko, “Geometry of Jacobi curves, I, II,” J. Dyn. Contr. Syst., 8, 93–140, 167–215 (2002).
V. I. Arnold and A. B. Givental, “Symplectic geometry,” in: Progress in Science and Technology, Series on Contemprorary Problems in Mathematics, Fundamental Direction [in Russian], 4, All-Union Institute for Scientific and Technical Information, USSR Acad. of Sciences, Moscow (1985), pp. 1–136.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton Univ. Press (1974).
I. Zelenko, “Variational approach to differential invariants of rank 2 vector distributions,” J. Diff. Geometry Appl. (in press).
Author information
Authors and Affiliations
Additional information
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.
Rights and permissions
About this article
Cite this article
Agrachev, A.A., Gamkrelidze, R.V. Vector fields on n-foliated 2n-dimensional manifolds. J Math Sci 135, 3093–3108 (2006). https://doi.org/10.1007/s10958-006-0147-1
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0147-1