Abstract:
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.
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