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4-dimensional anti-Kähler manifolds and Weyl curvature

Jaeman Kim (2006)

Czechoslovak Mathematical Journal

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.

A characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

Mayuko Kon (2008)

Czechoslovak Mathematical Journal

We give a characterization of totally η -umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M n ( c ) , c 0 , n 3 , satisfies g ( A X , Y ) = a g ( X , Y ) for any X , Y T 0 ( x ) , a being a function, where T 0 is the holomorphic distribution on M , then M is a totally η -umbilical real hypersurface or locally congruent to a ruled real hypersurface....

A complete classification of four-dimensional paraKähler Lie algebras

Giovanni Calvaruso (2015)

Complex Manifolds

We consider paraKähler Lie algebras, that is, even-dimensional Lie algebras g equipped with a pair (J, g), where J is a paracomplex structure and g a pseudo-Riemannian metric, such that the fundamental 2-form Ω(X, Y) = g(X, JY) is symplectic. A complete classification is obtained in dimension four.

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