On an existence theorem of Wagner manifolds

In their common paper [An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 43 (1997) 307–321] the authors give a condition for a 1-form β to be the perturbation of a Riemannian manifold (M, α) such that the manifold equipped with any (α, β) -metric is a Wagner manifold with respect to the Wagner connection induced by β. The condition shows that its covariant derivative with respect to the Lévi-Civita connection must be of a special form (∇β)(X,Y)=∥β#∥2α(X,Y)−β(X)β(Y)(∇β)(X,Y)=∥β#∥2α(X,Y)−β(X)β(Y)In this paper we give a family of Riemannian manifolds admitting nontrivial solutions and it will be proved as a structure theorem that there are no further essentially different examples; in particular we consider the classical hyperbolic space together with a nontrivial solution. Examples of Wagnerian Finsler manifolds are given in the last section.