Spherical twists, SO(n) and the lifting of their twist paths to Spin(n) in low dimensions

Let X ? Rn be a generalized annulus and consider the Dirichlet energy functional E[u;X] := 1/2?x|?u(x)|²dx, on the set of admissible maps A?(X) = {u?W²,¹(X, Sn¯¹): u|?X = ?}. Here ? ? C(?X, Sn¯¹) is fixed and A?(X) is non-empty. In this paper, we consider a class of maps referred to as spherical twists and examine them in connection with the Euler–Lagrange equation associated with E[·, X] on A?(X) (the harmonic map equation on X). The approach is novel and is based on lifting twist paths from SO(n) to its double cover Spin(n) and reformulating the harmonic map equation accordingly. We prove that, for n = 4 depending on ?, the system admits infinitely many smooth solutions in the form of twists or none, whereas, for n = 3 and in contrast, this number severely reduces to one or none.