Differential invariant

In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view. Differential invariants were introduced in special cases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. Lie (1884) was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equations, and invariant differential operators.

Differential invariants are contrasted with geometric invariants. Whereas differential invariants can involve a distinguished choice of independent variables (or a parameterization), geometric invariants do not. Élie Cartan's method of moving frames is a refinement that, while less general than Lie's methods of differential invariants, always yields invariants of the geometrical kind.

The simplest case is for differential invariants for one independent variable x and one dependent variable y. Let G be a Lie group acting on R². Then G also acts, locally, on the space of all graphs of the form y = ƒ(x). Roughly speaking, a k-th order differential invariant is a function

depending on y and its first k derivatives with respect to x, that is invariant under the action of the group.

The group can act on the higher-order derivatives in a nontrivial manner that requires computing the prolongation of the group action. The action of G on the first derivative, for instance, is such that the chain rule continues to hold: if

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