On Norden metrics which are locally conformal to anti-Kählerian metrics
A Norden manifold is a triplet \((M,J,g)\) where \(M\) is a smooth manifold, \(J\) is an almost complex structure on \(M\) (i.e., \(J^2= -id)\) and \(g\) is a necessarily pseudo-Riemannian metric of signature \((n,n)\) on \(M\) such that \(J\) becomes an anti-isometry at each point (i.e., \(g(JX,JY)= -g(X,Y)\) for all \(X,Y\in{\mathfrak X}(M)\)). A Norden structure \((J,g)\) is said to be anti-Kähler if \(\nabla J= 0\), where \(\nabla\) is the Levi-Civita connection of \(g\), and it is said to be locally conformally anti-Kähler if there exist local conformal deformations \(e^ug\) of \(g\) such that \((J,e^ug)\) becomes anti-Kähler on some open \({\mathfrak U}\subset M\). The authors' main concern is the construction of examples of locally conformally anti-Kähler structures on \(\Sigma\times S^1\), where \(\Sigma\) is either a pseudo-sphere of a pseudo-hyperbolic space, which are not anti-Kähler.
scientific article; zbMATH DE number 1506489