# zbMATH — the first resource for mathematics

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)
##### Keywords:
$$T$$-condition; Landsberg space
Full Text: