LEFT-INVARIANT PSEUDO-EINSTEIN METRICS ON LIE GROUPS

Abstract

In this article, we focus on left-invariant pseudo-Einstein metrics on Lie groups. To begin with, we give some examples of pseudo-Einstein metrics on Lie groups. Also we calculate the Levi-civita connection, and then Ricci tensor associated with left-invariant pseudo-Riemannian metrics on the unimodular Lie groups of dimension three. Furthermore, we show that the left-invariant pseudo-Einstein metric on SL(2) is unique up to a constant. At last, we study the left-invariant pseudo-Riemannian metrics on compact Lie groups and classify the pseudo-Einstein metrics on the low-dimensional compact Lie groups.