We know that conformal transformations of the Riemann sphere (which are general Mobius transformations) transform circles to circles. A similar fact is also true in higher dimensions. So we can say that circle is a concept in the conformal geometry of the spheres. Does there exist a more straightforward definition of circles in conformal geometry? and can one use this concept to give a more elementary proof of Liouville's theorem?

3$\begingroup$ See the paper of Bailey and Eastwood on conformal circles: ams.org/journals/proc/199010801/S00029939199009947717/… $\endgroup$ – Dan Fox Mar 9 '17 at 15:00
Conformal geometry is an example of Cartan geometry of parabolic type (aka parabolic geometry) and there is a notion of distinguished curves for any parabolic geometry. See e.g. On distinguished curves in parabolic geometries by A. Cap, J. Slovák, and V. Žádník or Parabolic geodesics as parallel curves in parabolic geometries by Marc Herzlich. This not only gives definition for higher dimensions but also for general curved conformal manifolds and other examples of parabolic geometries.