Abstract
We describe the (alpha ,beta )-metrics whose the T-tensor vanishes (T-condition) and the (alpha ,beta )-metrics that satisfy the sigma T-condition sigma _hT^h_{ijk}=0, where sigma _h=frac{partial sigma }{partial x^h} and sigma is a smooth function on M. These classes have already been obtained by Shen and Asanov in a completely different approach. The Finsler metrics of the first class are Berwaldian, the metrics of the second class are almost regular non-Berwaldian Landsberg metrics.
Highlights
The T -tensor plays an interesting role in Finsler geometry and general relativity
Hashiguchi [6] showed that a Landsberg space remains a Landsberg space under all conformal changes of the Finsler function if and only if its T -tensor vanishes
We calculate the T -tensor for the (α, β)-metrics, and we find necessary and sufficient conditions for (α, β)-metrics to satisfy the T -condition
Summary
The T -tensor plays an interesting role in Finsler geometry and general relativity. It was introduced by Matsumoto [9]. A Landsberg manifold remains of the same type under a conformal change (1.1) if and only if the T -tensor satisfies the condition σr. By solving some ODEs, we show that an (α, β)-metric satisfies the T -condition if and only if it is Riemannian or φ(s) has the following form φ (s ). We show that the (α, β)-metrics satisfy the σ T -condition if and only if the T -tensor vanishes (this is the trivial case) or φ(s) is given by φ(s) = c3 exp. Β α is a (positive definite) Finsler function if and only if φ satisfies the following conditions: φ(t) > 0, φ(t) − tφ (t) + (x2 − t2)φ (t) > 0,. The T -tensor of Randers metric has been studied by Matsumoto [8]
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