Generalization of the Chekanov theorem: diameters of immersed manifolds and wave fronts
Abstract
The Chekanov theorem generalizes the classic Lyusternik-Shnirel'man and Morse theorems concerning critical points of a smooth function on a closed manifold. A Legendrian submanifold \Lambda of space of 1-jets of the functions on a manifold M defines a multi-valued function whose graph is the projection of \Lambda in J^0 M = M x R. The Chekanov theorem asserts that if \Lambda is homotopic to the 1-jet of a smooth function in the class of embedded Legendrian manifolds, then such a graph of a multi-valued function must have a lot of points (their number is determined by the topology of M) at which the tangent plane to the graph is parallel to M \times 0. In the present paper a similar estimate is proved for a wider class of Legendrian manifolds. We consider Legendrian manifolds homotopic (in the class of embedded Legendrian manifolds) to Legendrian manifolds specified by generating families.
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Cite
@article{arxiv.math/9710219,
title = {Generalization of the Chekanov theorem: diameters of immersed manifolds and wave fronts},
author = {Petr E. Pushkar},
journal= {arXiv preprint arXiv:math/9710219},
year = {2016}
}