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Affine hyperspheres associated to special para-Kähler manifolds. (English) Zbl 1122.53006

In the paper a 1:1 correspondence between simply connected special para-Kähler manifolds \((M,J,g,\nabla)\) and improper affine hyperspheres with Blaschke structure \((M,g,\nabla)\) admitting a para-complex structure \(J\) such that \(\omega:=g(J.,.)\) is skew-symmetric and parallel is proved. The main idea of the proof is the construction of the conjugate (with respect to \(g\)) connection \(\nabla^J=J\circ\nabla\circ J\) which appears to be flat and torsionless and consequently determines together with \(\nabla\) and \(g\) a unique Blaschke immersion inducing \(\nabla\) and \(g\). There is also given a complete local description of affine improper para-special hyperspheres in terms of a non-degenerate para-holomorphic function. Finally, there is proved that every conical special para-Kähler manifold admits a foliation with leaves which are proper affine hyperspheres.

MSC:

53A15 Affine differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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