1 Introduction and main results

1. An important question of symplectic topology is the following: given a manifold, does it admit a Lagrangian embedding into Cn? A series of obstructions to existence of such embeddings arises due to pure topological reasons (see M. Audin's paper for a detailed discussion). However in M. Gromov discovered an obstruction of another nature. Using infinite-dimensional analysis he showed that on every embedded Lagrangian submanifold of Cn there exists a cycle with positive symplectic area. Thus the first Betti number of a manifold admitting a Lagrangian embedding into Cn does not vanish.

Besides the symplectic area there is another remarkable first cohomology class on Lagrangian submanifold of Cn – the Maslov class. Recently different restrictions on the Maslov class of Lagrangian embeddings were discovered. It is natural to suppose that they also lead to an obstruction to existence of Lagrangian embeddings. In the present paper we construct such obstruction (see theorem 1 below). We use it in order to show that certain flat manifolds do not admit Lagrangian embeddings into Cn (see theorems 2, 3 below). Our approach is based on the Maslov class rigidity phenomenon for manifolds of non-positive curvature which was discovered by C. Viterbo.

I am deeply grateful to I.H.E.S. for hospitality and to B. Bowditch, M. Gromov, M. Kapovich, J.-C. Sikorav and C. Vitero for numerous useful consultations and discussions.