On Ξ-curvature of m-th root Finsler metrics

The notions of S-curvature and Ξ-curvature introduced by Shen that is veryeffective for understanding the other Riemannian and non-Riemannian geometric propertiesof Finsler metrics. Here, we study the S-curvature and Ξ-curvature of the class of cubicand quartic (α, β)-metrics. We prove that third root (α, β)-metric of vanishing Ξ-curvaturereduces to a (−۱/۳)-Kropina metric or it has vanishing S-curvature. Then, we prove thatquartic (α, β)-metric of vanishing Ξ-curvature reduces to a special form of quartic (α, β)-metric or it has vanishing S-curvature.

کلیدواژه ها:

Ξ-curvature ، (α ، β)-metric ، S-curvature ، m–th root metric