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On isotropic submanifolds and evolution of quasicaustics. (English) Zbl 0806.58023
In previous work the author has studied coisotropic submanifolds and their generating families [Ann. Inst. Henri Poincaré, Phys. Théor. 56, No. 4, 429-441 (1992)]. In this paper the author applies the generating family approach to isotropic submanifolds. Recall that \(I\) is an isotropic submanifold of \((T^*M, \omega_ M)\) if \(i : I \to T^* M\) is an immersion and \(i^* \omega_ M = 0\). The quasicaustic of \(I\) is the image \(\pi_ M(I)\), where \(\pi_ M : T^ x M \to M\).
From the introduction: “In §1 we introduce the notion of \(I\)-Morse family generating an isotropic submanifold \(I\) and show geometric examples where isotropic submanifolds and their generating families appear naturally. In §2 we describe the general singularity theory machinery that can be used to classify isotropic submanifolds and their quasicaustics... In §3, [we give] the complete classification of generic evolutions of quasicaustic that can occur if \(\dim M < 4\)”.

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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