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Some special linear connection introduced in the Finsler space by Ichijy\=o has the property that the curvature tensors under conformal changes remain invariant. Two Ichijy\=o manifolds $(M,E,\nabla)$ and $(M,\overline E,\overline\nabla)$ are said to be conformally equivalent if $\overline E= (\exp\sigma^v)E$, $\sigma\in C^\infty(M)$.\par It is proved, that in this case, the following assertions are equivalent: 1. $\sigma$ is constant, 2. $h_\nabla= h_{\overline\nabla}$, 3. $S_{\nabla}= S_{\overline\nabla}$, 4. $t_\nabla= t_{\overline\nabla}$.\par It is also proved (when the above conditions are satisfied) that\par 1. If $(M,E,\nabla)$ is a generalized Berwald manifold, then $(M,\overline E,\overline\nabla)$ is also a generalized Berwald manifold.\par 2. If $(M,E,\nabla)$ is a Wagner manifold, then $(M,\overline E,\overline\nabla)$ is also a Wagner manifold with $\overline\alpha= \alpha+{1\over 2} \sigma$.\par A new proof of M. Hashiguchi's and Y. Ichijy\=o's theo! (English) |
