In mathematics, a diffiety () is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.
Intuitive definition
In algebraic geometry the main objects of study (varieties) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of polynomials), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraic ideal generated by the initial set of polynomials.
When dealing with differential equations, apart from applying algebraic operations as above, one has also the option to differentiate the starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal, one needs therefore to work with a differential ideal.
An elementary diffiety will consist therefore of the infinite prolongation of a differential equation , together with an extra structure provided by a special distribution. Elementary diffieties play the same role in the theory of differential equations as affine algebraic varieties do in the theory of algebraic equations. Accordingly, just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one defines a (non-elementary) diffiety as an object that locally looks like an elementary diffiety.
Formal definition
The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.
Jet spaces of submanifolds
For instance, for one recovers just points in and for one recovers the Grassmannian of -dimensional subspaces of . More generally, all the projections are fibre bundles.
As a particular case, when has a structure of fibred manifold over an -dimensional manifold , one can consider submanifolds of given by the graphs of local sections of . Then the notion of jet of submanifolds boils down to the standard notion of jet of sections, and the jet bundle turns out to be an open and dense subset of .
Prolongations of submanifolds
The -jet prolongation of a submanifold is
The map is a smooth embedding and its image , called the prolongation of the submanifold , is a submanifold of diffeomorphic to .
Cartan distribution on jet spaces
A space of the form , where is any submanifold of whose prolongation contains the point , is called an -plane (or jet plane, or Cartan plane) at . The Cartan distribution on the jet space is the distribution defined bywhere is the span of all -planes at .
Differential equations
A differential equation of order on the manifold is a submanifold ; a solution is defined to be an -dimensional submanifold such that . When is a fibred manifold over , one recovers the notion of partial differential equations on jet bundles and their solutions, which provide a coordinate-free way to describe the analogous notions of mathematical analysis. While jet bundles are enough to deal with many equations arising in geometry, jet spaces of submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such as Lagrangian submanifolds and minimal surfaces.
As in the jet bundle case, the Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to encode solutions in purely geometric terms. Indeed, a submanifold is a solution if and only if it is an integral manifold for , i.e. for all .
One can also look at the Cartan distribution of a PDE more intrinsically, definingIn this sense, the pair encodes the information about the solutions of the differential equation .