Handbook of Pseudo-Riemannian Geometry and Supersymmetry
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We study simply connected pseudo-Kähler and para-Kähler symmetric spaces that is simply connected pseudo-Riemannian symmetric spaces (M,g) which admit a parallel skew-symmetric complex or para-complex structure J ( such that J2 = −Id or J2 = Id respectively.) The classification of para-Kähler symmetric spaces reduces to description of 3-graded Lie algebras g = g−1 + g0 + g1 such that g0-modules g−1 and g1 are dual. The classification of pseudo-Kähler symmetric spaces reduces to description of some anti-involutive automorphisms of complex 3-graded Lie algebras. We classify para-Kähler and pseudo-Kähler Ricci-flat symmetric spaces of dimension ≤ 6. A class of Ricci-flat para-Kähler and pseudo-Kähler symmetric spaces with commutative holonomy group is described in terms of rank four tensors.
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