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

Abstract

For a C¹ function f:ℝ^n →ℝ\;(n \ge 2), we consider the least numberk of distinct critical points that f must possess when restricted 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 \cap V_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 by suitably restricted perturbations of quadratic forms having n distinct eigenvalues.