View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322201.png" /> (cf. [[Geometric objects, theory of|Geometric objects, theory of]]) be given in a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322202.png" />, the elements of which are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322203.png" />. A geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322204.png" /> of this manifold is known as a differential invariant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322206.png" /> with respect to the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322207.png" /> if its coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322209.png" />, are functions in the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222011.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222012.png" /> and their partial derivatives with respect to the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222014.png" />, up to the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222015.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222016.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222017.png" /></td> </tr></table>

the new coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222019.png" /> are expressed in terms of the new coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222021.png" /> and their partial derivatives with respect to the new coordinates by means of the same functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222022.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222023.png" /></td> </tr></table>

For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222024.png" /> be the object of a linear affine (torsion-free) connection. The object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222025.png" /> (curvature tensor)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222026.png" /></td> </tr></table>

is a tensor differential invariant of the first order with respect to the Christoffel symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222027.png" /> (cf. [[Christoffel symbol|Christoffel symbol]]).

Let there be given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222028.png" /> a group (pseudo-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222029.png" /> of point transformations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222031.png" /> be a submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222032.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222033.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

the parameters of which are subject to transformations of the infinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222035.png" />:

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A geometric differential invariant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222037.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222038.png" /> with respect to the group (pseudo-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222039.png" /> is the name of a function of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222040.png" /> of a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222041.png" /> and their partial derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222042.png" /> with respect to the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222043.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

which is invariant with respect to the transformations (1) and (2). In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222045.png" /> are substituted in (3) according to the formulas (1), while the partial derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222046.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222047.png" /> are replaced by their expressions in terms of derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222048.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222049.png" />, one obtains the same function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222051.png" /> and their derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222052.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222053.png" /></td> </tr></table>

If the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222054.png" /> are homogeneous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222055.png" /> should also be invariant with respect to the transformations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222056.png" /></td> </tr></table>

In the definition of a geometric differential invariant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222057.png" /> 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.Y. Thomas,   "The differential invariants of generalized spaces" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Weitzenböck,   "Invariantentheorie" , Noordhoff  (1923)</TD></TR></table>