Monge--Amp\'ere equations and characteristic connection functors

D. V. Tunitsky

Izvestiya. Mathematics , Tome 65 (2001) no. 6, pp. 1243-1290

Résumé

We investigate contact equivalence of Monge–Ampére equations. We define a category of Monge–Ampére equations and introduce the notion of a characteristic connection functor. This functor maps the category of Monge–Ampére equations to the category of affine connections. We give a constructive description of the characteristic connection functors corresponding to three subcategories, which include a large class of Monge–Ampére equations of elliptic and hyperbolic type. This essentially reduces the contact equivalence problem for Monge–Ampére equations in the cases under study to the equivalence problem for affine connections. Using E. Cartan's familiar theory, we are thus able to state and prove several criteria of contact equivalence for a large class of elliptic and hyperbolic Monge–Ampére equations.