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Matematicheskie Zametki, 2022, Volume 112, Issue 5, Pages 718–732
DOI: https://doi.org/10.4213/mzm13775
(Mi mzm13775)
 

This article is cited in 2 scientific papers (total in 2 papers)

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"

A. S. Mishchenkoab

a Lomonosov Moscow State University
b Moscow Center for Fundamental and Applied Mathematics
Full-text PDF (568 kB) Citations (2)
References:
Abstract: 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.
Received: 13.07.2022
English version:
Mathematical Notes, 2022, Volume 112, Issue 5, Pages 697–708
DOI: https://doi.org/10.1134/S0001434622110074
Bibliographic databases:
Document Type: Article
UDC: 51.73
Language: Russian
Citation: 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"”, Mat. Zametki, 112:5 (2022), 718–732; Math. Notes, 112:5 (2022), 697–708
Citation in format AMSBIB
\Bibitem{Mis22}
\by A.~S.~Mishchenko
\paper 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''
\jour Mat. Zametki
\yr 2022
\vol 112
\issue 5
\pages 718--732
\mathnet{http://mi.mathnet.ru/mzm13775}
\crossref{https://doi.org/10.4213/mzm13775}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4538801}
\transl
\jour Math. Notes
\yr 2022
\vol 112
\issue 5
\pages 697--708
\crossref{https://doi.org/10.1134/S0001434622110074}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85145364601}
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  • https://www.mathnet.ru/eng/mzm13775
  • https://doi.org/10.4213/mzm13775
  • https://www.mathnet.ru/eng/mzm/v112/i5/p718
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математические заметки Mathematical Notes
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