On caustics of submanifolds and canal hypersurfaces in Euclidean space (Q664660)

Consider an embedding \(x\) of an open set \(U\) of \(\mathbb R^r\) into \(\mathbb R^n\). There is a natural associated map, the canal hypersurface map \(y:U\times S^{n-r-1} \to \mathbb R^n\). For both of these maps \(x\) and \(y\), we can consider their distance-squared functions \(D_x\) and \(D_y\). The functions \(D_x\) and \(D_y\), considered as a ``Morse family of functions'', induce Lagrangian immersions \(L_x\) and \(L_y\). The first main theorem of the paper claims that the caustics of \(L_x\) and \(L_y\) coincide. The second main theorem claims that under some stability conditions, the induced Lagrangian immersion germs are Lagrangian equivalent. In the Appendix of the paper the authors construct `big' Legendrian submanifold germs and prove that every such object can be obtained by their construction. Natural equivalences of the inputs and the outputs of their construction are studied.