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Transactions of the American Mathematical Society

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The Maslov class of the Lagrange surfaces and Gromov's pseudo-holomorphic curves


Author: L. V. Polterovich
Journal: Trans. Amer. Math. Soc. 325 (1991), 241-248
MSC: Primary 58F05; Secondary 58G30
DOI: https://doi.org/10.1090/S0002-9947-1991-0992608-9
MathSciNet review: 992608
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Abstract: For an immersed Lagrange submanifold $ W \subset {T^\ast }X$, one can define a nonnegative integer topologic invariant $ m(W)$ such that the image of $ {H_1}(W;{\mathbf{Z}})$ under the Maslov class is equal to $ m(W) \cdot {\mathbf{Z}}$. In this paper, the value of $ m(W)$ is calculated for the case of a two-dimensional oriented manifold $ X$ with the universal cover homeomorphic to $ {{\mathbf{R}}^2}$ and an embedded Lagrange torus $ W$. It is proved that if $ X = {{\mathbf{T}}^2}$ and $ W$ is homologic to the zero section, then $ m(W) = 0$. In all the other cases $ m(W) = 2$. The last result is true also for a wide class of oriented properly embedded Lagrange surfaces in $ {T^\ast }{{\mathbf{R}}^2}$. The proof is based on the Gromov's theory of pseudo-holomorphic curves. Some applications to the hamiltonian mechanics are mentioned.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-0992608-9
Article copyright: © Copyright 1991 American Mathematical Society

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