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The symplectic structure of Kähler manifolds of non-positive curvature. (English) Zbl 0632.53058

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.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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