ON LEFT INVARIANT (alpha, beta)-METRICS ON SOME LIE GROUPS

摘要

We give the explicit formulas of the flag curvature of (alpha, beta)-metrics of Berwald type, and correct an error of the first and third authors in a previous article. Then we prove that at any point of a connected noncommutative nilpotent Lie group, the flag curvature of any left invariant (alpha, beta)-metric of Douglas type admits zero, positive and negative values, generalizing a theorem of Wolf. Moreover, we study left invariant (alpha, beta)-metrics of Douglas type on two interesting families of Lie groups considered by Milnor and Kaiser, including Heisenberg Lie groups. On these spaces, we present some necessary and sufficient conditions for (alpha, beta)-metrics to be of Berwald type, as well as some necessary and sufficient conditions for Randers metrics to be of Douglas type. We show that every left invariant (alpha, beta)-metric of Douglas type on G is an element of G(1), the family which is defined by Milnor, is a locally projectively flat Randers metric. We also give the explicit formulas of the flag curvature of left invariant Randers metrics of Douglas type on these spaces and show that, under a condition, the flag curvature is negative.