Monge--Amp\`ere equations and tensorial functors

D. V. Tunitsky

Izvestiya. Mathematics , Tome 73 (2009) no. 6, pp. 1217-1263

Résumé

We consider differential-geometric structures associated with Monge–Ampère equations on manifolds and use them to study the contact linearization of such equations. We also consider the category of Monge–Ampère equations (the morphisms are contact diffeomorphisms) and a number of subcategories. We are chiefly interested in subcategories of Monge–Ampère equations whose objects are locally contact equivalent to equations linear in the second derivatives (semilinear equations), linear in derivatives, almost linear, linear in the second derivatives and independent of the first derivatives, linear, linear and independent of the first derivatives, equations with constant coefficients or evolution equations. We construct a number of functors from the category of Monge–Ampère equations and from some of its subcategories to the category of tensorial objects (that is, multi-valued sections of tensor bundles). In particular, we construct a pseudo-Riemannian metric for every generic Monge–Ampère equation. These functors enable us to establish effectively verifiable criteria for a Monge–Ampère equation to belong to the subcategories listed above.

Détail

Keywords: Monge–Ampère equation, contact linearization, differential-geometric structures.