ON THE NUMBER OFCRITICAL POINTS OF A C¹ FUNCTION ON THE SPHERE

Abstract

For a C¹function f:ℝ^n →ℝ\;(n \ge 2), we consider the least numberk of distinct critical points that f must possess whenrestricted to the sphere S=\{x\in ℝ^n: \Vert x\Vert =1\}. Clearly k\ge 2 (for f attains its absolute minimum and maximum onS), and a result of Lusternik and Schnirelmann establishes thatk=n if f is even. Here we prove that k=n if,for a given orthonormal system (e_i), \max\limits_{S \capV_i}\,f<\min\limits_{S \cap V_i^\bot}\,f, for all i=1, …n-1,where V_i is the subspace spanned by e_1, …, e_i andV_i^\bot its orthogonal complement. It is shown that this criterion is satisfied bysuitably restricted perturbations of quadratic forms having n distincteigenvalues.