Theorems of Ljusternik-Schnirelmann

The Lusternik-Schnirelmann theorems relate to finding critical points of functions, particularly in topology and geometry. They establish that for certain shapes or spaces, the number of distinct critical points—points where a function's derivative is zero—is at least as many as the "category" of the space, a measure of its complexity. Essentially, these theorems guarantee that more complex topological structures cannot have too few critical points, ensuring multiple solutions or features exist in problems like optimization, curve analysis, or geometric configurations. They provide fundamental links between shape complexity and the presence of key points or solutions.