Geodesic
On the Geometry of Vector Fields
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Problems of Modern Topology and its Applications», Tome 144 (2018), pp. 81-87.

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It is well known that the study of the geometry and topology of the attainability set of a family of vector fields is one of the main tasks of the qualitative control theory, which is closely related to the geometry of orbits of vector fields. In this paper, we discuss the authors' results on the geometry of the attainability set of a family of vector fields: the results on the geometry of $T$-attainability sets and the geometry of orbits of Killing vector fields.
Keywords: vector field, attainability set, Killing vector field, Euler characteristic.
Mots-clés : orbit 
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A. Ya. Narmanov; S. S. Saitova. On the Geometry of Vector Fields. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Problems of Modern Topology and its Applications», Tome 144 (2018), pp. 81-87. http://geodesic.mathdoc.fr/item/INTO_2018_144_a8/

[1] Azamov A. A., Narmanov A. Ya., “O predelnykh mnozhestvakh orbit sistem vektornykh polei”, Differ. uravn., 40:2 (2004), 257–260 | MR | Zbl

[2] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, v. 1, 2, Nauka, M., 1981 | MR

[3] Narmanov A. Ya., Saitova S., “O geometrii orbit vektornykh polei Killinga”, Differ. uravn., 50:12 (2014), 1582–1589 | DOI | Zbl

[4] Narmanov A. Ya., Saitova S., “O geometrii mnozhestva dostizhimosti vektornykh polei”, Differ. uravn., 53:3 (2017), 321–326 | DOI | Zbl

[5] Narmanov A. Ya., Kasymov O., “O geometrii singulyarnykh rimanovykh sloenii”, Uzbek. mat. zh., 2011, no. 3, 113–121 | MR

[6] Narmanov A. Ya., Kasymov O., “O geometrii rimanovykh sloenii sfer malykh razmernostei”, Dokl. AN RUz, 2013, no. 2, 6–7

[7] Rashevskii P. K., “O soedinimosti lyubykh dvukh tochek vpolne negolonomnogo prostranstva dopustimoi liniei”, Uch. zap. Mosk. ped. in-ta im. K. Libknekhta. Ser. fiz.-mat. nauk, 1938, no. 2, 83–94

[8] Agrachev A. A., Sachkov Y., Control theory from the geometric viewpoint, Springer-Verlag, Berlin, 2004 | MR | Zbl

[9] Chow W. L., “Über Systeme von linearen partiellen Differentialgleinchungen ester Ordnung”, Math. Ann., 1939, no. 117, 98–105 | MR

[10] Hermann R., “The differential geometry of foliations”, J. Math. Mech., 2:11 (1962), 305–315 | MR

[11] R. Hermann, “A sufficient condition that a mapping of Riemannian manifolds be a fiber bundle”, Proc. Am. Math. Soc., 11 (1960), 236–242 | DOI | MR | Zbl

[12] Jurdjevich V., “Attainable sets and controllability: a geometry approach”, Lect. Notes Econ. Math. Syst., 106 (1974), 219–251 | DOI | MR

[13] Levitt N., Sussmann H., “On controllability by means of two vector fields”, SIAM J. Control., 13:6 (1975), 1271–1281 | DOI | MR | Zbl

[14] Nagano T., “Linear differential systems with singularities and application to transitive Lie algebras”, J. Math. Soc. Jpn., 18:4 (1968), 338–404 | MR

[15] O'Neil B., “The fundamental equations of a submersion”, Michigan Math. J., 13 (1966), 459–469 | DOI | MR | Zbl

[16] Stefan P., “Accessible sets, orbits, and foliations with singularities”, Proc. London Math. Soc., 29 (1974), 694–713 | MR

[17] Sussmann H., “Orbits of family of vector fields and integrability of systems with singularities”, Bull. Am. Math. Soc., 79 (1973), 197–199 | DOI | MR | Zbl