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Landsberg spaces satisfying the \(T\)-condition. (English) Zbl 0935.53015
If a Finsler space has a vanishing \(T\)-tensor: \(T_{ijkl}= C_{ijk}|_l+ L^{-1}(C_{ijk}|_l+C_{jkl}|_i +C_{ikl}|_j +C_{ijl}|_k)=0\), then it is said to satisfy the \(T\)-condition. If a Finsler space has vanishing \(hv\)-curvature tensor of the Cartan connection, then it is called a Landsberg space. The present paper is devoted to the study of \(n(\geq 3)\)-dimensional Landsberg spaces satisfying the \(T\)-condition. Some interesting results are obtained. For instance, if an \(n(\geq 3)\)-dimensional Landsberg space \(M^n\) satisfying the \(T\)-condition is \(S3\)-like, that is, \(S_{ijkl}= S(h_{ik} h_{jl}-h_{il} h_{jk})\) and \(S\) is not equal to \(-L^{-2}\), then \(M^n\) is conformally flat if and only if \(M^n\) is locally a Minkowski space.
MSC:
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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