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On the conformal change of Douglas space of second kind with special (α, β)-metric

The notion of Douglas space of the second kind of the Finsler space with (α, β)-metric was introduced by I. Y. Lee [1], Since then, many of the geometers have studied this topic [2] [14], In this paper, we prove that the Douglas space of second kind with special (α,β)-metric ([17] [18] [19]) α+β+kβ2α is conformally transformed to a Douglas space of second kind. Further, we obtain some results which prove that the Douglas space of second kind with certain (α,β)-metrics such as Randers metric, special (α,β)-metric α+β2α, first approximate Matsumoto metric and Finsler space with square metric is conformally transformed to a Douglas space of second kind. In [3], S. Bacso and M. Matsumoto developed the concept of Douglas space as an extension of Berwald space from view point of geodesic equations. Also, they considered the concept of Landsberg space as extension of Berwald space. In [4], S. Bacso and B. Szilagyi introduced the concept of weakly-Berwald space as another extension of Berwald space. In this paper, we introduce Douglas space of second kind and study the condition under which it is conformally invariant.