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 $ \Omega $( cf. Geometric objects, theory of) be given in a differentiable manifold $ X _ {n} $, the elements of which are points $ ( u ^ {1} \dots u ^ {n} ) $. A geometric object $ \omega $ of this manifold is known as a differential invariant of order $ r $ with respect to the object $ \Omega $ if its coordinates $ \omega _ {A} $, $ A = 1 \dots N $, are functions in the coordinates $ \Omega _ \alpha $, $ \alpha = 1 \dots M $, of $ \Omega $ and their partial derivatives with respect to the coordinates $ u ^ {i} $, $ i = 1 \dots n $, up to the order $ r $:

$$ \omega _ {A} = f _ {A} ( \Omega _ \alpha , \partial _ {i} \Omega _ \alpha \dots \partial _ {i _ {1} \dots i _ {r} } ^ {r} \Omega _ \alpha ), $$

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

$$ u ^ {i} = u ^ {i} ( u ^ {1 ^ \prime } \dots u ^ {n ^ \prime } ) , $$

the new coordinates $ \omega _ {A} ^ \prime $ of $ \omega $ are expressed in terms of the new coordinates $ \Omega _ {A} ^ \prime $ of $ \Omega $ and their partial derivatives with respect to the new coordinates by means of the same functions $ f _ {A} $:

$$ \omega _ {A} ^ \prime = f _ {A} ( \Omega _ {A} ^ \prime , \partial _ {i} \Omega _ \alpha \dots \partial _ {i _ {1} ^ \prime \dots i _ {n} ^ \prime } ^ {r} \Omega _ \alpha ) . $$

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

$$ R _ {ijk} ^ { l } = \partial _ {i} \Gamma _ {jk} ^ {l} - \partial _ {j} \Gamma _ {ik} ^ {l} + \Gamma _ {is} ^ {l} \Gamma _ {jk} ^ {s} - \Gamma _ {js} ^ {l} \Gamma _ {ik} ^ {s} $$

is a tensor differential invariant of the first order with respect to the Christoffel symbols $ \Gamma _ {ij} ^ {k} $( cf. Christoffel symbol).

Let there be given in $ X _ {n} $ a group (pseudo-group) $ G $ of point transformations

$$ \tag{1 } u ^ {i} = f ^ { i } ( \overline{\mathbf u}\; {} ^ {1} \dots \overline{\mathbf u}\; {} ^ {n} ) $$

and let $ M _ {h} $ be a submanifold of $ X _ {n} $ of dimension $ h $:

$$ \tag{2 } u ^ {i} = \phi ^ {i} ( t ^ {1} \dots t ^ {n} ) , $$

the parameters of which are subject to transformations of the infinite group $ G _ \infty $:

$$ t ^ \alpha = \psi ^ \alpha ( \overline{t}\; {} ^ {1*} \dots \overline{t}\; {} ^ {n*} ) . $$

A geometric differential invariant of order $ r $ of the manifold $ M _ {h} $ with respect to the group (pseudo-group) $ G $ is the name of a function of the coordinates $ u ^ {i} $ of a point of $ M _ {h} $ and their partial derivatives up to order $ r $ with respect to the parameters $ t ^ \alpha $:

$$ \tag{3 } F \left ( u ^ {i} , \frac{\partial u ^ {i} }{\partial t ^ \alpha } \dots \frac{\partial ^ {r} u ^ {i} }{\partial t ^ {\alpha _ {1} } {} \dots \partial t ^ {\alpha _ {r} } } \right ) , $$

which is invariant with respect to the transformations (1) and (2). In fact, if $ u ^ {i} $ are substituted in (3) according to the formulas (1), while the partial derivatives of $ u ^ {i} $ with respect to $ t ^ \alpha $ are replaced by their expressions in terms of derivatives of $ \overline{\mathbf u}\; {} ^ {i} $ with respect to $ \overline{\mathbf t}\; {} ^ {\alpha ^ {*} } $, one obtains the same function $ F $ in $ \overline{\mathbf u}\; {} ^ {i} $ and their derivatives with respect to $ \overline{\mathbf t}\; {} ^ {\alpha ^ {*} } $:

$$ F \left ( u ^ {i} , \frac{\partial {\overline{\mathbf u}\; } {} ^ {i} }{\partial {\overline{\mathbf t}\; } {} ^ {\alpha _ {1} } } \dots \frac{\partial ^ {r} {\overline{\mathbf u}\; } {} ^ {i} }{\partial {\overline{\mathbf t}\; } {} ^ {\alpha _ {r} } \dots \partial {\overline{\mathbf t}\; } {} ^ {\alpha _ {r} } } \right ) . $$

If the coordinates $ u ^ {i} $ are homogeneous, then $ F $ should also be invariant with respect to the transformations

$$ u ^ {* _ {i} } = \lambda ( t ^ {1} \dots t ^ {n} ) u ^ {i} ,\ \ \lambda \neq 0 . $$

In the definition of a geometric differential invariant, $ F $ 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)