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.
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References
A. Cannas da Silva, Lectures on Symplectic Geometry, in Lecture Notes in Math. (Springer, Berlin, 2008), Vol. 1764.
V. V. Trofimov and A. T. Fomenko, Algebra and Geometry of Integrable Hamiltonian Differential Equations (Faktorial, Moscow, 1995) [in Russian].
V. A. Vasil’ev, Lagrange and Legendre Characteristich Classes (MTsNMO, Moscow, 2000) [in Russian].
A. T. Fomenko, Symplectic Geometry (Izd. Moskov. Univ., 1988) [in Russian].
V. I. Arnol’d, “Characteristic class entering in quantization conditions,” Funct. Anal. Appl. 1 (1), 1–13 (1967).
D. B. Fuchs, “Maslov–Arnol’d characteristic classes,” Sov. Math. Dokl. 9, 96–99 (1968).
Acknowledgments
This research was initiated by discussions with A. T. Fomenko and V. E. Nazaikinskii, to both of whom I express my deep gratitude.
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Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 718–732 https://doi.org/10.4213/mzm13775.
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Mishchenko, A.S. 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”. Math Notes 112, 697–708 (2022). https://doi.org/10.1134/S0001434622110074
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DOI: https://doi.org/10.1134/S0001434622110074