Symplectic twistor spaces.(English)Zbl 0810.53056

Summary: We introduce some canonical 2-form in the twistor bundles of any Riemannian manifold $$M$$. This form is always closed and turns out to be nondegenerate in the following cases: 1. The curvature of $$M$$ is pinched. 2. $$M$$ is an Einstein four-dimensional manifold of positive or negative curvature. 3. $$M$$ is self-dual and the Ricci curvature is pinched.
We prove the existence of a large class of compact symplectic manifolds which do not admit any Kählerian structure. Finally, we introduce a Lagrangian lift of a totally geodesic submanifold of $$M$$ and apply the Lagrangian intersection methods to totally geodesic submanifolds.

MSC:

 53C56 Other complex differential geometry 53C20 Global Riemannian geometry, including pinching
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