-spaces of Iwasawa type and algebraic rank one.
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.
We give a characterization of totally -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 of a real hypersurface of a complex space form , , , satisfies for any , being a function, where is the holomorphic distribution on , then is a totally -umbilical real hypersurface or locally congruent to a ruled real hypersurface....
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.
A holomorphic representation formula for special parabolic hyperspheres is given.