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Special Finsler spaces admitting metric like tensor field. (English) Zbl 1173.53036

This paper studies special Finsler spaces such as \(C\)-reducible, semi-\(C\)-reducible, quasi-\(C\)-reducible with the \(T\)-condition which are admitting a tensor field \(X_{hk}=h_{hk}+X_{00}l_hl_k\) that satisfies the condition \(C^h_{ij} X_{hk}=C_{ijk}\). But any concrete example of such tensor field \(X_{hk}\) is not found up to the present. There is a wrong situation.
The authors use the notions of semi-\(C\)-reducible [M. Matsumoto and S. Numata, Tensor, New Ser. 34, 218–222 (1980; Zbl 0435.53023)] and \(T\)-condition [H. Rund, The differential geometry of Finsler spaces. Heidelberg: Springer-Verlag (1959; Zbl 0087.36604)] without dimension restriction. It is also remarked that the \(T\)-condition was first defined in [M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces. Shiga-Ken 520, Japan: Kaiseisha Press (1986; Zbl 0594.53001), p.189] and it should be noted that \(l_ml^m=l\).

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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