Weakly symmetry of a class of g-natural metrics on tangent bundles

Abstract

The paper considers a class of g-natural metrics G on the tangent bundle TM of a Riemannian manifold (M, g). We prove that the flatness of g is necessary and sufficient for a metric G to be weakly symmetric (recurrent or pseudo-symmetric). Also, it is shown that the weak symmetry and recurrent or pseudo-symmetry properties of Sasakian lift metric, studied by Bejan and Crasmareanu, and Binh and Tamássy, respectively, are special cases of our result.