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On a theorem of Chekanov. (English) Zbl 0893.53015

Budzyński, Robert (ed.) et al., Symplectic singularities and geometry of gauge fields. Proceedings of the Banach Center symposium on differential geometry and mathematical physics in Spring 1995, Warsaw, Poland. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 39, 39-48 (1997).
The author discusses from a geometrical point of view a proof of the following theorem of Chekanov. Denote by \(\lambda_0\) and \(\lambda_1\) the extremities of a Legendrian isotopy of the space of 1-jets of functions. If \(\lambda_0\) is induced by a generating family quadratic at infinity (g.f.q.i.), so is \(\lambda_1\). In particular, any quasi-function is induced by some g.f.q.i.
The author proves a similar result in the context of spherized cotangent bundles. Let \(l_0\) and \(l_1\) be the extremities of a Legendrian isotopy of the spherized cotangent bundle. If \(l_0\) has a good generating hypersurface, then so has \(l_1\). In particular, any generalized quasi-function has a good generating hypersurface.
The author also generalizes the theorem of the number of critical points of quasi-functions on a compact closed manifold in the setting of the projectivized cotangent bundle and presents a construction based on the hodograph transform which yields some interesting consequences on the extrinsic geometry of wave fronts.
For the entire collection see [Zbl 0863.00038].

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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