SBC work only for real-valued functions, i.e. functions that map the parameter space to the real numbers, which can be ordered and hence admit well-defined ranks. Any function into the real numbers will give valid ranks and hence a valid SBC.

Typically our ambient space is \mathbb{R}^{N} and we can use the global coordinate functions for SBC. For an N-sphere, however, there aren’t coordinate functions that span the entire space and hence not immediate candidates. Indeed a map from, say, a point on a circle to a circular angle is not real-valued and hence isn’t applicable, which is why one can’t work out self-consistent results.

There are, however, many other valid functions that one could use instead. For example, one could use one of the infinite the maps from a circle into the real interval [0, 2 \pi]. In other words one can use angle coordinate functions (like \theta for a circle or \theta, \phi for a 2-sphere) and treat the discontinuity as infinity.