On a property of functions on the sphere and its application

Given a continuous function f:Sm+n−2→Rmf:Sm+n−2→Rm, and nn points u1,u2,…,un∈Sm+n−2u1,u2,…,un∈Sm+n−2; does there exist a rotation r∈SO(m+n−1)r∈SO(m+n−1) such that f(ru1)=f(ru2)=⋯=f(run)f(ru1)=f(ru2)=⋯=f(run)? In this paper, we study the property of a continuous map from a sphere to a Euclidean space by using the theory of Smith periodic transformation and Brouwer degree of map theorem. The conjecture is proved under the case of n=2n=2 and mm being even. Furthermore, this conjecture is proved for the case when uj⋅uj+1=λuj⋅uj+1=λ and the dimension of the sphere is not less than m+n−2m+n−2. This paper generalizes the Borsuk–Ulam theorem and then presents its application.