Differential invariant

An expression composed of one or more functions, their partial derivatives of various orders with respect to independent variables, and sometimes also the differentials of these variables, which is invariant with respect to certain transformations.

Let a geometric object (cf. Geometric objects, theory of) be given in a differentiable manifold , the elements of which are points . A geometric object of this manifold is known as a differential invariant of order with respect to the object if its coordinates , , are functions in the coordinates , , of and their partial derivatives with respect to the coordinates , , up to the order :

and have the following property of invariance with respect to coordinate transformations. In fact, under a change of coordinates

the new coordinates of are expressed in terms of the new coordinates of and their partial derivatives with respect to the new coordinates by means of the same functions :

For instance, let be the object of a linear affine (torsion-free) connection. The object (curvature tensor)

is a tensor differential invariant of the first order with respect to the Christoffel symbols (cf. Christoffel symbol).

Let there be given in a group (pseudo-group) of point transformations

(1)

and let be a submanifold of of dimension :

(2)

the parameters of which are subject to transformations of the infinite group :

A geometric differential invariant of order of the manifold with respect to the group (pseudo-group) is the name of a function of the coordinates of a point of and their partial derivatives up to order with respect to the parameters :

(3)

which is invariant with respect to the transformations (1) and (2). In fact, if are substituted in (3) according to the formulas (1), while the partial derivatives of with respect to are replaced by their expressions in terms of derivatives of with respect to , one obtains the same function in and their derivatives with respect to :

If the coordinates are homogeneous, then should also be invariant with respect to the transformations

In the definition of a geometric differential invariant, may be replaced by a geometric object. If this object is a covariant (contravariant) vector, it is named covariant (contravariant).

If the vanishing of some object is invariant, the object is named a relative differential invariant.

References

[1]	T.Y. Thomas, "The differential invariants of generalized spaces" , Cambridge Univ. Press (1934)
[2]	R. Weitzenböck, "Invariantentheorie" , Noordhoff (1923)