arXiv subjects
Publications and source records attributed to Petr E. Pushkar.
Main subject of the paper is a (strong) Morse function on a compact manifold with boundary. We construct a cellular structure and discuss its algebraic properties in this paper. Also we get an estimation on Arnold's question on a number of critical points of a Morse function with given boundary condition.
We prove a Chekanov-type theorem for the spherization of the cotangent bundle $ST^*B$ of a closed manifold $B$. It claims that for Legendrian submanifolds in $ST^*B$ the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies.
We construct counterexamples to lifting properties of Hamiltonian and contact isotopies.
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 Λof space of 1-jets of the functions on a manifold M defines a multi-valued function whose graph is the projection of Λin J^0 M = M x R. The Chekanov theorem asserts that if Λis 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.