Examples of solvmanifolds without certain affine structure (Q1320938)

The authors discuss the conjecture of \textit{J. Milnor} (1977) claiming that any simply connected solvable Lie group \(G\) admits a complete left- invariant affinely flat structure. The positive answer for nilpotent groups \(G\) was published by \textit{Boyom} (1989). In the reviewed paper, a Lie group \(G=N \rtimes \mathbb{R}^ 4\), where \(N= \mathbb{R}^{13}\) is its nilradical, is constructed, in such a way that the following condition is satisfied. There is no embedding \(\theta: G\to \text{Aff}(17)\) such that \(\theta(N)\subset \mathbb{R}^{17}\) inducing an action of \(S/N= \mathbb{R}^ 4\) on \(\mathbb{R}^{17}/ \mathbb{R}^{13}= \mathbb{R}^ 4\) by translations.