On the left invariant $(\alpha,\beta)$-metrics on some Lie groups

TL;DR

This paper derives explicit formulas for flag curvatures of certain invariant Finsler metrics on Lie groups, showing their curvature can be positive, negative, or zero, and characterizes conditions for these metrics to be of Berwald or Douglas type.

Contribution

It provides new explicit formulas for flag curvatures of invariant Matsumoto and Kropina metrics, and characterizes conditions for Berwald and Douglas types on specific Lie groups.

Findings

Flag curvature of invariant metrics can be positive, negative, or zero.

Explicit formulas for flag curvatures of Matsumoto and Kropina metrics are obtained.

Conditions for metrics to be of Berwald or Douglas type are established.

Abstract

We give the explicit formulas of the flag curvatures of left invariant Matsumoto and Kropina metrics of Berwald type. We can see these formulas are different from previous results given recently. Using these formulas, we prove that at any point of an arbitrary connected non-commutative nilpotent Lie group, the flag curvature of any left invariant Matsumoto and Kropina metrics of Berwald type admits zero, positive and negative values, this is a generalization of Wolf's theorem. Then we study $(\alpha,\beta)$-metrics of Berwald type and also Randers metrics of Douglas type on two interesting families of Lie groups considered by Milnor and Kaiser, containing Heisenberg Lie groups. On these spaces, we present some necessary and sufficient conditions for $(\alpha,\beta)$-metrics to be of Berwald type and also some necessary and sufficient conditions for Randers metrics to be of Douglas type. All left invariant non-Berwaldian Randers metrics of Douglas type are given and the flag curvatures are computed.