Submanifolds of Kähler manifolds of non-positive curvature

After Gromov's discovery of the existence of exotic symplectic structures on R2n one important problem has been the understanding of the standard symplectic structure itself. McDuff proved a global version of the Darboux Theorem which states that

Theorem 1.1The Kähler form ω on a simply connected complete Kähler 2n-dimensional manifold P of non-positive sectional curvature is diffeomorphic to the standard symplectic form ω0 on R2n.

This means in particular that the symplectic structure on a Hermitian symmetric space of non-compact type is standard. She also showed that

Theorem 1.2If L is a totally geodesic connected properly embedded Lagrangian submanifold of such a manifold P, then P is symplectomorphic to the cotangent bundle T*L with its usual symplectic structure.

Recall that a submanifold Q of P is said to be symplectic if ω restricts to a symplectic form on Q and is said to be isotropic if the restriction of ω to Q is identically zero. In the complex hyperbolic space CHn of complex dimension n, the complex hyperbolic subspaces CHi, 0 ≤ i ≤ n, are examples of totally geodesic symplectic submanifolds and the real hyperbolic subspaces Hn−i, 0 ≤ i ≤ n, are examples of totally geodesic isotropic submanifolds.

Throughout this section we assume that Q is a totally geodesic connected properly embedded submanifold of (P, ω).