We study closed 1-forms with isolated zeros on closed oriented surfaces and establish the conditions under which given invariants generate a closed 1-form.
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S. Kh. Aranson and V. Z. Grines, “Topological classification of the flows on closed surfaces,” Usp. Mat. Nauk, 41, Issue 1(247), 149–169 (1986).
S. Kh. Aranson, V. Z. Grines, and E. V. Zhuzhoma, “Using the Lobachevsky plane to study surfaces and 2-webs flows, foliation,” in: Proc. Internat. Conf. BGL-4, (September, 7–11, 2004, Nizhny Novgorod; Kiev) (2004), pp. 8–24.
S. Kh. Aranson, E. V. Zhuzhoma, and V. S. Medvedev, “On the continuity of the geodesic frames of flows on the surfaces,”Mat. Sb., 188, No. 7, 3–22 (1997).
S. Kh. Aranson and V. Z. Grines, “On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive systems),” Mat. Sb., 90(132), No. 3, 372–401 (1973).
S. V. Bilun and O. O. Pryshlyak, “Closed Morse 1-forms on closed surfaces,” Visn. Kyiv Univ., Ser. Mat. Mekh., No. 8, 77–81 (2002).
N. V. Budnyts’ka and O. O. Pryshlyak, “Equivalence of closed 1-forms on oriented surfaces,” Visn. Kyiv Univ., Ser. Mat. Mekh., No. 19, 36–38 (2008).
N. V. Budnyts’ka, “Equivalence of closed 1-forms on closed nonoriented surfaces,” Nelin. Kolyv., 12, No. 2, 155–167 (2009); English translation: Nonlin. Oscillations, 12, No. 2, 157–170 (2009).
N. V. Budnyts’ka and O. O. Pryshlyak, “Equivalence of closed 1-forms on the surfaces with edge,” Ukr. Mat. Zh., 61, No. 11, 1455–1472 (2009); English translation: Ukr. Math. J., 61, No. 11, 1710–1727 (2009).
N. V. Budnyts’ka, “Realization of a closed 1-form with closed recurrent curves on closed surfaces,” in: Problems of Topology and Related Problems. Proceedings of the Institute of Mathematics of the Ukrainian National Academy of Sciences [in Ukrainian], 6, No. 2, 340–348 (2009).
A. O. Prishlyak, “Topological equivalence of smooth functions with isolated critical points on a closed surface,” Topol. Appl., No. 119, 257–267 (2002).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 6, pp. 739–751, June, 2012.
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Budnyts’ka, N.V., Rybalkina, T.V. Realization of a closed 1-form on closed oriented surfaces. Ukr Math J 64, 844–856 (2012). https://doi.org/10.1007/s11253-012-0683-4
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DOI: https://doi.org/10.1007/s11253-012-0683-4