Unicorn metrics with almost vanishing ${\bf H}$- and ${\bf \Xi}$-curvatures

Abstract

In this paper, we consider a class of almost regular $(\alpha, \beta)$-metrics constructed by Shen called unicorn metrics. First, we prove that every unicorn metric with almost vanishing ${\bf H}$-curvature is a Berwald metric. Then we show that every unicorn metric with almost vanishing $\Xi$-curvature reduces to a Berwald metric.