New classification method for Equivalence Classes on $S^1\mathrm{/}{Z}_2$ and $T^2\mathrm{/}{Z}_3$ Orbifolds

In five- and six-dimensional $U(N)$ and $SU(N)$ gauge theories compactified on $S^1\mathrm{/}{Z}_2$ and $T^2\mathrm{/}{Z}_3$ orbifolds, we propose a new method to classify the equivalence classes (ECs) of boundary conditions (BCs) wihtout depending on the structure of gauge transformations. Some of the BCs are connected through gauge transformations and constitute ECs, each of which contains physically equivalent BCs. Previous methods for classifying ECs have been used specific gauge transformations. In this paper, we show that a geometric property of orbifolds significantly narrows down the possibilities of connecting BCs and completes the classification of ECs.