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

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.