McDuff, Dusa The symplectic structure of Kähler manifolds of non-positive curvature. (English) Zbl 0632.53058 J. Differ. Geom. 28, No. 3, 467-475 (1988). 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. Cited in 2 ReviewsCited in 17 Documents MSC: 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:Kähler form; Kähler manifold; symplectic form; Hermitian symmetric space; Lagrangian submanifold; comparison theorem PDFBibTeX XMLCite \textit{D. McDuff}, J. Differ. Geom. 28, No. 3, 467--475 (1988; Zbl 0632.53058) Full Text: DOI