We consider relations between two families of flat manifolds with holonomy group (Z_2)^k of diagonal type. The family ${ \cal RBM}$ of real Bott manifolds and the family ${ \cal GHW}$ of generalized Hantzsche-Wendt manifolds. In particular, we prove that the intersection ${ \cal GHW}\cap { \cal RBM}$ is not empty. We also consider some class of real Bott manifolds without $\operatorname{ Spin}$ and $\operatorname{ Spin}^{ \C}$ structure. There are given conditions for the (non)existence of such structures.
Categories: math.AT math.GT