Conformal geodesics are the critical points of the conformal arclength functional on curves in the conformal n-sphere, viewed as a homogeneous space under the Moebius group. Writing their Euler-Lagrange equations in every dimension by frame reduction rather than by Griffiths' formalism shows that the span of the first few frame vectors together with the invariant normal field and its covariant derivative is a Lorentzian subspace that the equations force to be constant along the curve. Since that subspace has dimension at most six, every conformal geodesic lies in a totally umbilical conformal 4-sphere, so the problem in any dimension reduces to dimension four. The dimension of the constant subspace then splits the solutions into constant-curvature, three-degenerate and three-generic cases, and the three-generic system of curvature equations is integrated by elliptic functions, yielding an explicit expression up to conformal motion for exactly those conformal geodesics that lie in no conformal 3-sphere.
The conformal sphere Q^n is the n-sphere presented as the Darboux hyperquadric in projective space, on which every conformal diffeomorphism acts linearly, making it a homogeneous space under the Moeb…