S-Curvature of left invariant Randers metrics on some simple Lie groups

In this paper we study the Riemannian geometry of simple Lie groups SO(۳,R), SL(۲,R) and SO(۱,۳), equipped with a left invariant Riemannian metric. We consider left invariant Randers metrics induced by these left invariant Riemannian metrics. Then, in each case, we obtain the S-curvature and show that although these Randers metrics are not of Berwald or Douglas type but in the case of SO(۳,R) it is of almost isotropic S-curvature. Finally, we give the S-curvature of left invariant Randers metrics on four-dimensional Einstein Lie groups.

Keywords:

S-curvature , isotropic S-curvature , Lie groups , Randers space