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 Nikvcević's results [6, Theorems 3.1 and 4.1].

We classify all the closed 3‐dimensional orbifolds with Sol‐geometry. These are aspherical orbifolds and so their fundamental groups determine the orbifolds completely. Thus we will classify all the crystallographic groups of Sol, together with all the Bieberbach groups, up to isomorphism.

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Abstract. Suppose that S and S are simply connected solvable Lie groups of type (R) with the same dimension. We show that the Lefschetz coincidence numbers of maps f, g : Γ\S → Γ \S between special solvmanifolds Γ\S → Γ \S can be computed algebraically as follows:where F * , G * are the matrices, with respect to any preferred bases, of morphisms of Lie algebras induced by f and g. This generalizes a recent result by S. W. Kim and J. B. Lee to special solvmanifolds of type (R). Moreover, we can drop the dimension match condition imposed in the latter result.

The authors gave an example showing an error in [2, Lemma 3.3], and below offer at least a partial correction for that error under the unimodularity assumption. This makes all of the remaining results in [2] valid.Consider the three-dimensional solvable non-unimodular Lie algebra S: