The exponential deformation retract of the sphere \(S^n\) and its folding (Q1818392)

An isometric folding \(F: M\to N\) between Riemannian manifolds maps piecewise geodesic paths of~\(M\) to piecewise geodesic paths of~\(N\) of the same length. In this paper, the author considers a deformation retract of \(S^n\setminus \{p,q\}\) onto~\(S^{n-1}\), where \(p,q\) are two conjugate points of~\(S^n\). This deformation retract is lifted to~\(T_pS^n\) via the exponential map. The author shows that the lifted deformation behaves well under an isometric folding of~\(S^n\) and the induced isometric folding of the tangent space~\(T_pS^n\).