Critical points of quasi-functions and generating families of Legendrian manifolds (Q677681)

The known Lyusternik-Schnirelmann and Morse theorems provide the lower bound for the number of critical points of a smooth function on a closed manifold \(M\). In the interpretation of these results as statements concerning the number of points of intersection of an exact Lagrangian section of the cotangent bundle \(T^*M\) with the zero section, this estimate remains valid if the exact Lagrangian section is replaced by an arbitrary exact embedded Lagrangian manifold homotopic to the zero section [see \textit{M. Chaperon}, C. R. Acad. Sci., Paris, Sér. I 298, 293-296 (1984; Zbl 0576.58010) and \textit{F. Laudenbach} and \textit{J.-C. Sikorav}, Invent. Math. 82, 349-357 (1985; Zbl 0592.58023)]. In the reviewing article the further generalization of these results is given, where Legendrian submanifolds of the bundle of 1-jets occur instead of exact Lagrangian submanifolds of the cotangent bundle. In this generalization the basic role plays the theorem asserting that the subset of manifolds determined by generating families quadratic at infinity is open and closed in the \(C^\infty\) topology of the set of all embedded Legendrian submanifolds.