Nucleon spin decomposition and differential geometry

In the last few years, the so-called Chen et al. approach of the nucleon spin decomposition has been widely discussed and elaborated on. In this letter we propose a genuine differential geometric understanding of this approach. We mainly highligth its relation to the "dressing field method" we advocated in [C. Fournel, J. Fran\c{c}ois, S. Lazzarini, T. Masson, Int. J. Geom. Methods Mod. Phys. 11, 1450016 (2014)]. We are led to the conclusion that the claimed gauge-invariance of the Chen et al. decomposition is actually unreal.