Compact flat spacetimes (Q1886838)

The author gives a precise classification of compact flat space-times. He uses different techniques from those ones used in the paper by \textit{F. Grunewald} and \textit{G. Margulis} [J. Geom. Phys. 5, 493--531 (1988; Zbl 0706.57022)]. By a purely geometric approach he proves that a compact flat space-time (up to a finite cover) is either a \(K\)-step nilmanifold with \(k\leq 3\), or a 2-step solvmanifold diffeomorphic to the product \(T^m \times T^{2k+3}_A\), where \(n=m+2k+3\) and \(A\) is a Lorentz matrix in SL\(_{2k+3}(\mathbb{Z})\). Then he uses this result to study the existence of closed time-like and null geodesics in compact flat space-times.