Round Morse functions and isoenergy surfaces of integrable Hamiltonian systems
Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 319-336 Cet article a éte moissonné depuis la source Math-Net.Ru

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A study is made of the topology of a class of three-dimensional manifolds that arise as constant energy surfaces for integrable systems. It is proved that this class coincides with the class of manifolds admitting a function all of whose critical manifolds are nondegenerate circles and whose nonsingular level surfaces are disjoint unions of tori. Necessary and sufficient conditions are obtained for the existence of minimal round Morse functions on manifolds of dimension greater than five. Figures: 3. Bibliography: 20 titles.
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     author = {S. V. Matveev and A. T. Fomenko and V. V. Sharko},
     title = {Round {Morse} functions and isoenergy surfaces of integrable {Hamiltonian} systems},
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S. V. Matveev; A. T. Fomenko; V. V. Sharko. Round Morse functions and isoenergy surfaces of integrable Hamiltonian systems. Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 319-336. http://geodesic.mathdoc.fr/item/SM_1989_63_2_a3/

[1] Fomenko A. T., “Topologiya poverkhnostei postoyannoi energii integriruemykh sistem i prepyatstviya k integriruemosti”, Izv. AN SSSR, 50 (1986), 1276–1307 | MR | Zbl

[2] Fomenko A. T., “Topologiya trekhmernykh mnogoobrazii i integriruemye mekhanicheskie sistemy”, V Tiraspolskii simp. po obschei topologii i ee prilozheniyam, Shtiintsa, Kishinev, 1985, 235–237

[3] Fomenko A. T., “Teoriya Morsa integriruemykh gamiltonovykh sistem”, DAN SSSR, 287:5 (1986), 1071–1075 | MR | Zbl

[4] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[5] Novikov S. P., “Variatsionnye metody i periodicheskie resheniya uravnenii tipa Kirkhgofa. II”, Funktsion. analiz i ego pril., 15:4 (1981), 37–52 | MR | Zbl

[6] Trofimov V. V., Fomenko A. T., “Integriruemost po Liuvillyu gamiltonovykh sistem na algebrakh Li”, UMN, 39:2 (1984), 3–56 | MR | Zbl

[7] Matveev S. V., Burmistrova A. B., “Struktura $S$-funktsii na orientiruemykh 3-mnogoobraziyakh”, Tez. XI Vsesoyuzn. shkoly po teorii operatorov v funktsionalnykh prostranstvakh, Izd-vo ChPI, Chelyabinsk, 1986, 12 | MR

[8] Morgan J. W., “Non-singular Morse-Smale flows on 3-dimensional manifolds”, Topology, 18:1 (1979), 41–53 | DOI | MR | Zbl

[9] Yokoyoma Kazuo, “On links with property $P^*$”, Yokohama Math. J., 25 (1977), 71–84 | MR

[10] Waldhausen F., “Eine Klasse von 3-dimensionalen Mannigfaltigleiten. I, II”, Invent Math., 3:4 (1967), 308–333 | DOI | MR | Zbl

[11] Matveev S. V., Savvateev V. V., “Trekhmernye mnogoobraziya, imeyuschie prostye spetsialnye ostovy”, Soll. Math., XXXII:2 (1974), 83–97

[12] Thurston W., “Existence of codimension-one foliations”, Ann. Math., 104:2 (1976), 249–268 | DOI | MR | Zbl

[13] Miyosihi S., “Foliated round surgery of codimension-one foliated manifolds”, Topology, 21:3 (1982), 245–262 | DOI | MR

[14] Asimov D., “Round handles and non-singular Morse-Smale flows”, Ann. Math., 102:1 (1975), 41–54 | DOI | MR | Zbl

[15] Smale S., “On the structure of manifolds”, Amer. J. Math., 84:3 (1962), 387–399 | DOI | MR | Zbl

[16] Fomenko A. T., Topologicheskie variatsionnye zadachi, izd-vo MGU, M., 1984 | MR | Zbl

[17] Oshemkov A. A., “Bottovskie integraly nekotorykh integriruemykh gamiltonovykh sistem”, Teometriya, differentsialnye uravneniya i mekhanika, Izd-vo MGU, M., 1986, 115–117 | MR

[18] Franks J., “The periodic structure of non-singular Morse–Smale flows”, Comment. Math. Helv., 53:2 (1978), 279–294 | DOI | MR | Zbl

[19] Fomenko A. T., Zieschang H., On the topology of threedimensional manifolds arising in Hamiltonian mechanics, Preprint. Bericht No 55/1985, Universität Bochum, Ruhr, 1985

[20] Trofimov V. V., Fomenko A. T., “Geometriya skobok Puassona i metody integrirovaniya po Liuvillyu sistem na simmetricheskikh prostranstvakh”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 29, VINITI, M., 1986, 3–108 | MR