On the number of critical points of a C1 function on the sphere

For a C1 function f:Rn →R (n≥2) , we consider the least number k of distinct critical points that f must possess when restricted to the sphere S={x∈Rn :∥x∥=1} . Clearly k≥2 (for f attains its absolute minimum and maximum on S), and a result of Lyusternik and Shnirelʹman establishes that k=n if f is even. Here we prove that k=n if, for a given orthonormal system (ei ), max S∩Vi f