= Lusternik–Schnirelmann critical point theorem
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= LS critical point theorem
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Every smooth real-valued function on a closed connected <smooth manifold> has at least its unnormalized <Lusternik–Schnirelmann category> many distinct <critical points>. No nondegeneracy hypothesis is required. If there are infinitely many critical points the bound is immediate; in the finite case, the sublevel-set deformation argument shows that category can increase only at critical levels, by at most the number of critical points on that level. In conjunction with <cup length>, this gives at least $r+1$ critical points whenever there is a nonzero $r$-fold positive-degree <cup product>. This theorem is distinct from the antipodal covering theorem also sometimes named the Lusternik–Schnirelmann theorem. A primary account of the critical-point formulation is https://doi.org/10.1016/j.exmath.2017.10.003[Conley pairs in geometry—Lusternik–Schnirelmann theory and more].
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