Finsler space with the general approximate Matsumoto metric (Q1865982)

A special \((\alpha,\beta)\)-metric \(L=\alpha^2/(\alpha -\beta)\) is called a Matsumoto metric. This is expressed as \[ L= \lim_{r\to\infty} \alpha \sum^r_{k=0} (\beta/ \alpha)^k, \] for \(|\beta |< |\alpha |\), and approximated by \(L=\alpha \sum^r_{k=0} (\beta/ \alpha)^k\). If \(r=0\), then \(L= \alpha\) is a Riemannian metric. If \(r=2\), then \(L=\alpha+\beta\) is a Randers metric. The Finsler space considered in the present paper is equipped with an approximate metric, called the general approximate Matsumoto metric. On the other hand, a Finsler space is called a Douglas space, if its Douglas tensor vanishes identically [\textit{M. Matsumoto}, Tensor, New Ser. 60, 123-134 (1998)]. The main result of the present paper is the following Theorem: If a Finsler space of dimension greater than two with a general approximate Matsumoto metric is a Douglas space, then it it a Berwald space.