Some mapping theorems for continuous functions defined on the sphere

According to the Knaster conjecture, for any continuous function f:Sm+n−2→Rm and n distinct points u1,u2,…,un∈Sm+n−2, there exists a rotation r∈SO(m+n−1) such that f(ru1)=f(ru2)=⋯=f(run). In this paper, we focus on the study of the properties of a continuous mapping from a sphere to a Euclidean space by using the theory of Smith periodic transformation and the Smith special index of the Stiefel manifold under periodic transformation. We obtain some mapping theorems for the case where n=αβ,β is an odd prime number and ui⋅uj=ui+α⋅uj+α(1≤i,j≤n,un+1=u1). Furthermore, if n=p, where p is an odd prime number, this conjecture is proved for the case where ui⋅uj=ui+1⋅uj+1(1≤i,j≤p,up+1=u1) and the dimension of the sphere is not less than m+n−2.