Maslov form and J-volume of totally real immersions

On every totally real submanifold Mⁿ of ⁿ, one can define a Maslov class analogous to the one defined for the Lagrangian submanifolds of ⁿ. We define here a closed 1-form, expressed in terms of the extrinsic local geometric invariants of Mⁿ and the complex structure of ⁿ, whose cohomology class is the Maslov class of Mⁿ. This generalizes to the totally real case, the result of Morvan (1981). This 1-form can still be defined if the ambient space ⁿ is substituted by a Kahler manifold , but it is not closed in general. However, we can build a variational problem on the space of totally real immersions, whose critical points are totally real submanifolds whose form defined above vanishes identically. In the case where , we give a characterization and many examples of such submanifolds. Finally we study the second variation and prove a stability result for the critical submanifolds of a Kahler manifold with non-positive Ricci tensor. This extends the well-known results on Lagrangian submanifolds of