Left invariant metrics and curvatures on simply connected three‐dimensional Lie groups

Abstract

Abstract For each simply connected three‐dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the principal Ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three‐dimensional Lie groups. Our results improve a bit of Milnor's results of [7] in the three‐dimensional case, and Kowalski and Nikv́cević's results [6, Theorems 3.1 and 4.1] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)