Notes About a New Metric on the Cotangent Bundle

Abstract

In this article, we construct a new metric $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over G} = {}^R\nabla + \sum\limits_{i,j = 1}^m {a^{ji}} \delta {p_j}\delta {p_i}$ in the cotangent bundle, where ${}^R\nabla $ is the Riemannian extension and $ a^{ji}$ is a symmetric (2,0)-tensor field on a differentiable manifold.