The circle $S^1$ can be thought of as the coset space of:
- Special orthogonal groups, $S^1 \simeq SO(2)/SO(1)$,
- Orthogonal groups, $S^1 \simeq O(2)/O(1)$.
Which other groups is the circle a coset space for?
Thinking about $S^1$ as a classic configuration space (from a physicist's perspective), which salient features might incline us to represent our circle using one group over another? Of course each is valid mathematically, but it seems there should be some reason we might pick say (1) over (2).
Wikipedia's entry for homogeneous spaces seems to suggest (1) yields the oriented sphere whereas (2) does not. What does that mean precisely?