Geodesic
Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $W=\mathbb{T}^*(M)$ of the Cotangent Bundle”
Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 718-732

Voir la notice de l'article provenant de la source Math-Net.Ru

The geometric properties of the Maslov index on symplectic manifolds are discussed. The Maslov index is constructed as a homological invariant on a Lagrangian submanifold of a symplectic manifold. In the simplest case, a Lagrangian submanifold $\Lambda\subset \mathbb{R}^{2n}\approx \mathbb{R}^{n}\oplus\mathbb{R}^{n}$ is a submanifold in the symplectic space $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$, in which the symplectic structure is given by the nondegenerate form $\omega=\sum_{i=1}^n dx^{i}\wedge dy^{i}$ and $\Lambda\subset\mathbb{R}^{2n}$ is a submanifold, $\dim\Lambda=n$, on which the form $\omega$ is trivial. In the general case, a symplectic manifold $(W,\omega)$ and the bundle of Lagrangian Grassmannians $\mathcal{LG}(\mathbb{T}W)$ is considered. The question under study is as follows: when is the Maslov index, given on an individual Lagrangian manifold as a one-dimensional cohomology class, the image of a one-dimensional cohomology class of the total space $\mathcal{LG}(\mathbb{T}W)$ of bundles of Lagrangian Grassmannians? An answer is given for various classes of bundles of Lagrangian Grassmannians.
Keywords: Maslov index, Maslov class, symplectic manifold, bundle of Lagrangian manifolds. 
A. S. Mishchenko. Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $W=\mathbb{T}^*(M)$ of the Cotangent Bundle”. Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 718-732. http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a6/
@article{MZM_2022_112_5_a6,
     author = {A. S. Mishchenko},
     title = {Maslov {Index} on {Symplectic} {Manifolds.} {With} {Supplement} by {A.~T.~Fomenko} {{\textquotedblleft}Constructing} the {Generalized} {Maslov} {Class} for the {Total} {Space} $W=\mathbb{T}^*(M)$ of the {Cotangent} {Bundle{\textquotedblright}}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {718--732},
     year = {2022},
     volume = {112},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a6/}
}
TY  - JOUR
AU  - A. S. Mishchenko
TI  - Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $W=\mathbb{T}^*(M)$ of the Cotangent Bundle”
JO  - Matematičeskie zametki
PY  - 2022
SP  - 718
EP  - 732
VL  - 112
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a6/
LA  - ru
ID  - MZM_2022_112_5_a6
ER  - 
%0 Journal Article
%A A. S. Mishchenko
%T Maslov Index on Symplectic Manifolds. With Supplement by A. T. Fomenko “Constructing the Generalized Maslov Class for the Total Space $W=\mathbb{T}^*(M)$ of the Cotangent Bundle”
%J Matematičeskie zametki
%D 2022
%P 718-732
%V 112
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a6/
%G ru
%F MZM_2022_112_5_a6

[1] A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Math., 1764, Springer, Berlin, 2008 | DOI | MR

[2] V. V. Trofimov, A. T. Fomenko, Algebra i geometriya integriruemykh gamiltonovykh dtfferentstalnykh uravnenii, Faktorial, M., 1995 | MR

[3] V. A. Vasilev, Lagranzhevy i lezhandrovy kharakteristicheskie klassy, MTsNMO, M., 2000

[4] A. T. Fomenko, Simplekicheskaya geometriya, Izd-vo Mosk. un-ta, 1988

[5] V. I. Arnold, “O kharakteristicheskom klasse, vkhodyaschem v usloviya kvantovaniya”, Funkts. analiz i ego pril., 1:1 (1967), 1–14 | MR | Zbl

[6] D. B. Fuks, “O kharakteristicheskikh klassakh Maslova–Arnolda”, Dokl. AN SSSR, 178:2 (1968), 303–306 | MR | Zbl