The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a transitive group of isometries are obtained. These conditions are Intrinsic, Deductive, Explicit and ALgorithmic, and they offer an IDEAL labeling of these geometries. It is shown that the transitive action of the group naturally falls into an unfolding of some of the ten types in the Bianchi-Behr classification. Explicit conditions, depending on the Ricci tensor, are obtained that characterize all these types.

中文翻译：

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齐次三维黎曼空间

得到了三维黎曼度量允许传递等距群的充要条件。这些条件是内在的、演绎的、显式的和算法的，它们为这些几何提供了理想的标记。结果表明，群的传递行为自然地属于 Bianchi-Behr 分类中十种类型中某些类型的展开。根据 Ricci 张量，获得表征所有这些类型的显式条件。