The analysis in gauge space

Based on the analogy and interplay between rotations in ordinary space and gauge space, generalized two-dimension bands Esub(α)(N,I) and the crossing of them have been used for analyzing the back bending phenomena in both spaces simultaneously. The shape transition and shape coexistence as a function of N and I are discussed in detail for rare earth nuclei, Hg isotopes, and Kr isotopes. The N and I dependence of Z=64 shell closure is analyzed. And some hint from N(lambda,I) plot for pairing disappearance is discussed preliminarily. Finally the differences between two spaces are mentioned. (orig.)