In this article, we construct a new metric 3pt = R∇ + ∑i,j = 1ᵐ aʲⁱ δ pⱼδ pᵢ in the cotangent bundle, where R∇ is the Riemannian extension and aʲⁱ is a symmetric (2,0)-tensor field on a differentiable manifold.
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Filiz Ocak (2019) studied this question.
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