The symplectic structure of Kähler manifolds of non-positive curvature (Q1095420)

In this note we show that the Kähler form on a simply connected complete Kähler manifold W of non-positive curvature is diffeomorphic to the standard symplectic form on \({\mathbb{R}}^ n\). This means in particular that the symplectic structure on a Hermitian symmetric space of non-compact type is standard. We also show that if L is a totally geodesic proper connected Lagrangian submanifold of a complete Kähler manifold W of non-positive curvature then W is symplectomorphic to the cotangent bundle T*L with its usual symplectic structure provided that \(\pi _ 1(W,L)=0\). The proofs use a comparison theorem due to Greene-Wu and Siu-Yau.