Here, we deals with a special lift ˜ ???? of a Riemannian metric ???? on a manifold ???? to the tangent bundle ???????? of ????. This lift is defined as a linear combination of certain well-known lifts of ????. The main results of the paper are proved under the condition that the Riemannian manifold (???? ????) is flat, in fact the Riemannian connection of the metric ˜ ???? coincides with the complete lift of the Riemannian connection of the metric ????. In addition, the main objectives of this study is to find the necessary and sufficient conditions such that some of the lift vector fields with this general metric to be parallel.

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