I was about to write a similar story, but then I realized you need a Riemannian metric (and exponential) for everything arclength-related. IIRC since the unit

I was about to write a similar story, but then I realized you need a Riemannian metric (and exponential) for everything arclength-related. IIRC since the unit circle is a compact Lie group, there is a bi-invariant Riemannian metric whose exponential is the Lie group exponential and we land back on our feet, but the Lie group structure alone is not sufficient.