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

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 Nikvcevic's results [6, Theorems 3.1 and 4. 1]. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim