Lagrangian submanifolds in para-Kähler manifolds

In this paper we initiate the study of Lagrangian submanifolds in para-Kähler manifolds. In particular, we prove two general optimal inequalities for Lagrangian submanifolds of the flat para-Kähler manifold (E2n,g̃,P). Moreover, we completely classify Lagrangian submanifolds which satisfy the equality case of one of the two inequalities.