Derivatives with respect to horizontal and vertical lifts of the modified Riemannian extension g˜∇,c on Cotangent Bundle

In this paper, we define the modified Riemannian extension g˜∇,c in the cotangent bundle T ∗M, which is completely determined by its action on vector fields of type X H and ω V . Later, we obtain the covarient and Lie derivatives applied to the modified Riemannian extension with respect to the horizontal and vertical lifts of vector and kovector fields, respectively