A new class of Finsler metrics: Douglas curvature and its generalizations

This paper introduces new classes of Finsler metrics, namely ?D-stretch metrics, isotropic ?D- stretch metrics, and relatively isotropic ?D-metrics, by exploring the Douglas curvature in Finsler geometry. The class of relatively isotropic ?D-metrics encompasses two additional classes: ?D-stretch metrics and isotropic ?D-stretch metrics. The study delves into the properties of relatively isotropic ?D-metrics, elucidating their geometric characteristics and situating them within the broader context of Finsler metrics dependent on Douglas curvature, such as Douglas or GDW-metrics. Additionally, the research investigates the interplay between relatively isotropic ?D-metrics and other key curvatures, including ?E-curvature and S-curvature, building upon prior studies on the relationships between Douglas curvature and these curvatures. Furthermore, examples of Finsler metrics are provided to elucidate the distinguishing criteria for the class of relatively isotropic ?D-metrics in comparison to well-known classes of Finsler metrics like Douglas, Weyl, and GDW-metrics.