Flat pseudo-Riemannian Lie groups
A result of \textit{J. W. Milnor} [Adv. Math. 21, 293--329 (1976; Zbl 0341.53030)] describes the structure of Lie groups which carry a flat left-invariant Riemannian metric. The authors are interested in the generalization of this study to the pseudo-Riemannian situation. In the non-definite case, the group is not necessarily complete with respect to the metric, a fact which turns out to be equivalent to the group being unimodular. The authors give methods to construct examples, showing in particular that the skew-cotangent group of an affine group always carries a flat invariant metric with signature \((n,n)\). They finally show that some groups, like the general linear group or the Heisenberg groups of dimension \(\geq 5\) cannot carry such a metric.
scientific article; zbMATH DE number 2091872