Geodesic
Geometry and topology of two-dimensional symplectic manifolds with generic singularities and Hamiltonian systems on them
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2024), pp. 22-33 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The topological and symplectic classifications of closed 2-dimensional symplectic manifolds whose symplectic structure has generic singularities are obtained. The Liouville foliations of Hamiltonian systems on such manifolds are classified in topological category. The properties of index-one surgery along a pair of Liouville tori are studied together with the singularities of symplectic structure it gives rise to. The change of Liouville foliation topology after the surgery in dimension two is described. 
@article{VMUMM_2024_5_a2,
     author = {A. Yu. Konyaev and E. A. Kudryavtseva and V. I. Sidel'nikov},
     title = {Geometry and topology of two-dimensional symplectic manifolds with generic singularities and {Hamiltonian} systems on them},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {22--33},
     year = {2024},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2024_5_a2/}
}
TY  - JOUR
AU  - A. Yu. Konyaev
AU  - E. A. Kudryavtseva
AU  - V. I. Sidel'nikov
TI  - Geometry and topology of two-dimensional symplectic manifolds with generic singularities and Hamiltonian systems on them
JO  - Vestnik Moskovskogo universiteta. Matematika, mehanika
PY  - 2024
SP  - 22
EP  - 33
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/VMUMM_2024_5_a2/
LA  - ru
ID  - VMUMM_2024_5_a2
ER  - 
%0 Journal Article
%A A. Yu. Konyaev
%A E. A. Kudryavtseva
%A V. I. Sidel'nikov
%T Geometry and topology of two-dimensional symplectic manifolds with generic singularities and Hamiltonian systems on them
%J Vestnik Moskovskogo universiteta. Matematika, mehanika
%D 2024
%P 22-33
%N 5
%U http://geodesic.mathdoc.fr/item/VMUMM_2024_5_a2/
%G ru
%F VMUMM_2024_5_a2
A. Yu. Konyaev; E. A. Kudryavtseva; V. I. Sidel'nikov. Geometry and topology of two-dimensional symplectic manifolds with generic singularities and Hamiltonian systems on them. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2024), pp. 22-33. http://geodesic.mathdoc.fr/item/VMUMM_2024_5_a2/

[1] Martinet J., “Sur les singularités des formes différentielles”, Annales de l'institut Fourier, 20:1 (1970), 95–178 | DOI

[2] Zotev D.B., “O simplekticheskoi geometrii mnogoobrazii s pochti vsyudu nevyrozhdennoi zamknutoi 2-formoi”, Matem. zametki, 76:1 (2004), 66–77 | DOI

[3] Zotev D.B., “Kontaktnye vyrozhdeniya zamknutykh 2-form”, Matem. sb., 198:4 (2007), 47–78 | DOI

[4] Zotev D.B., “Predkvantovanie po Kostantu simplekticheskikh mnogoobrazii s kontaktnymi osobennostyami”, Matem. zametki, 105:6 (2019), 857–878 | DOI

[5] Cardona R., Miranda E., “On the volume elements of a manifold with transverse zeroes”, Regul. Chaot. Dyn., 24 (2019), 187–197 | DOI

[6] Moser J., “On the volume elements on a manifold”, Trans. Amer. Math. Soc., 120 (1965), 286–294 | DOI

[7] Fomenko A.T., “Topologiya poverkhnostei postoyannoi energii integriruemykh gamiltonovykh sistem i prepyatstviya k integriruemosti”, Izv. AN SSSR. Ser. matem., 50:6 (1986), 1276–1307

[8] Fomenko A.T., “Simplekticheskaya topologiya vpolne integriruemykh gamiltonovykh sistem”, Uspekhi matem. nauk, 44:1 (265) (1989), 145–173

[9] Fomenko A.T., Tsishang Kh., “Topologicheskii invariant i kriterii ekvivalentnosti integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody”, Izv. AN SSSR. Ser. matem., 54:3 (1990), 546–575

[10] Bolsinov A.V., Fomenko A.T., Integriruemye gamiltonovy sistemy. Topologiya. Geometriya. Klassifikatsiya, Udmurtskii universitet, Izhevsk, 1999

[11] Zotev D.B., Sidelnikov V.I., “Realizatsiya invariantov Fomenko–Tsishanga v zamknutykh simplekticheskikh mnogoobraziyakh s kontaktnymi osobennostyami”, Matem. sb., 213:4 (2022), 3–26 | DOI

[12] Milnor J., Lectures on the $h$-Cobordism Theorem, Princeton Univ. Press, Princeton, New Jersey, 1965

[13] Moser J., “Regularization of Kepler's problem and the averaging method on a manifold”, Comm. Pure and Appl. Math., 23:4 (1970), 609–636 | DOI

[14] Levi-Civita T., “Sur la résolution qualitative du problème restreint des trois corps”, Verhandlungen des dritten Internationalen Mathematiker-Kongresses in Heidelberg vom 8. bis 13. August 1904, 1905, Leipzig, 402–408

[15] Levi-Civita T., “Sur la résolution qualitative du problème restreint des trois corps”, Acta Math., 30:1 (1906), 305–327 | DOI

[16] Kudryavtseva E.A., Lepskii T.A., “Topologiya lagranzhevykh sloenii integriruemykh sistem s giperellipticheskim gamiltonianom”, Matem. sb., 202:3 (2011), 69–106 | DOI

[17] Kudryavtseva E.A., “Analog teoremy Liuvillya dlya integriruemykh gamiltonovykh sistem s nepolnymi potokami”, Dokl. RAN, 445:4 (2012), 383–385

[18] Fomenko A.T., “Teoriya bordizmov integriruemykh gamiltonovykh nevyrozhdennykh sistem s dvumya stepenyami svobody. Novyi topologicheskii invariant mnogomernykh integriruemykh sistem”, Izv. AN SSSR. Ser. matem., 55:4 (1991), 747–779