Unified classification of pure metric geometries

Abstract

Almost Norden, almost product Riemannian, almost Norden golden and almost golden Riemannian are pure metric geometries. We introduce $\alpha$-metric and $\alpha$-golden metric manifolds to unify the study of almost Norden manifolds and almost product Riemannian manifolds with null trace and almost Norden golden manifolds and almost golden Riemannian manifolds with null trace respectively. Then we can show the classifications of almost Norden manifolds and almost product Riemannian manifolds with null trace in a unified way. The bijection between $\alpha$-metric and $\alpha$-golden metric manifolds allows us to classify $\alpha$-golden metric manifolds, i.e., we classify almost Norden golden manifolds and almost golden Riemannian manifolds with null trace simultaneously. Finally we characterize every class of the above four kind of pure metric manifolds by means of the first canonical and the well-adapted connections which are two distinguished connections shared by $\alpha$-metric and $\alpha$-golden metric manifolds.