ИСТИНА

The definition of the Maslov index of a Lagrangian manifold as a class of one-dimensional cohomology on it has given rise to numerous works gen- eralizing the concepts of the Maslov index. In the works of V.I.Arnold, V.A.Vasiliev and their followers, the theory of Lagrangian bordisms was developed and characteristic classes of Lagrangian submanifolds were con- structed on its basis. There is another approach to describing Maslov classes of Lagrangian submanifolds, presented in the works of V.V.Trofimov and A.T.Fomenko from a categorical point of view. In addition, there is a series of works by M.V.Karasev and V.P.Maslov in which the canonical operator is constructed for general symplectic manifolds by covering a symplectic manifold with charts diffeomorphic to the simplest symplectic manifold. Inspired by the works of V.V.Trofimov and A.T.Fomenko, we introduce the concept of so-called infinitesimal Lagrangian manifolds, which, in our opinion, allow us to characterize the characteristic classes of Lagrangian manifolds with maximum completeness and calculate the Maslov index for almost any Lagrangian manifolds.