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.^[1] 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. The moving frames method, which is a refinement of Élie Cartan's method of moving frames, gives several new powerful tools for finding and classifying the equivalence and symmetry properties of submanifolds, differential invariants, and their syzygies. Although, the moving frames method is less general than Lie's methods of differential invariants, it always yields invariants of the geometrical kind.