On the Conformal change of a Douglas space of second kind with special $(\alpha, \beta )$-metric

TL;DR

This paper investigates how Douglas spaces of second kind with specific $(eta)$-metrics, including Randers and Kropina, are preserved under conformal transformations, extending previous geometric results in Finsler geometry.

Contribution

It proves that Douglas spaces of second kind with certain $(eta)$-metrics remain Douglas spaces under conformal changes, including special and well-known metrics.

Findings

Conformal transformation preserves Douglas space of second kind for the specified $(eta)$-metrics.

Results include Randers, Kropina, Matsumoto, and square metrics.

Provides new conditions under which these spaces remain Douglas spaces.

Abstract

The notion of a Douglas space of second kind of a Finsler space with $(\alpha, \beta)$-metric was introduced by I. Y. Lee [9]. Since then, so many geometers have studied this topic e. g., [14]. In this paper, we prove that a Douglas space of second kind with special $% (\alpha, \beta)$-metric $\alpha +\epsilon \beta + k \frac{\beta^2}{\alpha }$ is conformally transformed to a Douglas space of second kind. Further, we obtain some results which prove that a Douglas space of second kind with certain $(\alpha, \beta)$-metrics such as Randers metric, Kropina metric, first approximate Matsumoto metric and Finsler space with square metric is conformally transformed to a Douglas space of second kind.