MASLOV FORM AND J-VOLUME OF TOTALLY-REAL IMMERSIONS

On every totally real submanifold M-n of C-n, one can define a Maslov class analogous to the one defined for the Lagrangian submanifolds of C-n. We define here a closed 1-form, expressed in terms of the extrins ic local geometric invariants of Mn and the complex structure of Cn, w hose cohomology class is the Maslov class of M-n. This generalizes to the totally real case, the result of Morvan (1981). This 1-form can st ill be defined if the ambient space C-n is substituted by a Kahler man ifold (M) over tilde(2n), but it is not closed in general. However, we can build a variational problem on the space of totally real immersio ns, whose critical points are totally real submanifolds whose form def ined above vanishes identically In the case where (M) over tilde(2n) = C-n, we give a characterization and many examples of such submanifold s. Finally we study the second variation and prove a stability result for the critical submanifolds of a Kahler manifold with non-positive R icci tenser. This extends the well-known results on Lagrangian submani folds of (M) over tilde(2n). (C) 1998 Elsevier Science B.V.