Lie groups as four-dimensional complex manifolds with Norden metric (Q1021300)

An even dimensional almost complex manifold \((M, J, g)\), with Norden metric, say \(G\), has the usual almost complex metric structure \((J, g)\) such that \(G(X, Y) = g(X, JY)\) for all \(X, Y\) on \(M\) and both metrics are indefinite of signature \((n, n)\). Moreover, \((M, G)\) is called an isotropic Kähler manifold if \(\| \nabla J\|^2 = 0\) where \(\nabla\) is the Levi-Civita connection of the metric \(g\). In this paper, the authors have constructed two examples of \(4\)-dimensional complex manifolds with Norden metric using Lie groups and Lie algebras. Finally, they give geometric conditions for such manifolds to be isotropic Kählerian.