View source for Geometric objects, theory of

A branch of differential geometry based on the theory of group representations. The use of the method of exterior differential forms makes it possible to introduce differential criteria into the theory of geometric objects, which convert it to an effective tool in differential-geometric studies of spaces with fundamental groups, as well as of generalized spaces (fibre spaces, spaces with a connection, differentiable manifolds endowed with different differential-geometric structures).

Let each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442701.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442702.png" />-parameter Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442703.png" /> be put into correspondence with a transformation of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442704.png" /> belonging to some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442705.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442706.png" />, and let the zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442707.png" /> of the group correspond to the identity transformation (mapping) of the space into itself. Let a successive execution of transformations by two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g0442709.png" /> be equivalent with the transformation effected by the product of these elements, and let a coordinate system be suitably introduced into the space. One then says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427010.png" /> is locally represented on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427011.png" /> as a transformation group. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427012.png" /> is called a representation space of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427013.png" /> or a space with fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427014.png" />.

A geometric object with given fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427015.png" />, or a geometric object associated to a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427016.png" /> (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427018.png" />-object for short), is defined as a point of the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427019.png" />. This space itself is called a space of geometric objects in the wide sense of the word, or a generalized homogeneous space. A group of transformations of a space of geometric objects which realizes its fundamental group is called the fundamental group of this geometric object. Two geometric objects in the same representation space of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427020.png" /> are called equivalent if one of them may be transformed into the other by a transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427021.png" />. An intransitivity system is called a space of geometric objects in the proper sense. A representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427022.png" /> is called a homogeneous space with fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427023.png" /> if on it a faithful transitive representation of this group is realized. In any faithful representation space of a finite group there exists a frame consisting of a finite number of points.

Let a faithful transitive representation be realized in the space of geometric objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427024.png" />. Let the representation space be a manifold and together with a frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427025.png" /> let it be subjected to all possible transformations of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427026.png" />, then a complete family (space) of frames is obtained on which a simply-transitive representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427027.png" /> is realized. This space is identified with the group space, or the parameter space, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427028.png" />. If an arbitrary point (frame) of this space is taken as reference point and is brought into correspondence with the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427029.png" />, then all the points of this space are brought in a one-to-one correspondence with the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427030.png" />. Group parameters may be regarded as parameters of the moving frame.

It is also possible to establish a one-to-one correspondence between the frames of the family and the elements of the group such that each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427031.png" /> of the group is brought into correspondence with the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427032.png" /> obtained from an arbitrary fixed initial (absolute) frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427033.png" /> by right (left) shift by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427034.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427035.png" />. To a current element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427036.png" /> will correspond a current  "moving"  frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427037.png" />. Each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427038.png" /> of the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427039.png" /> is defined with respect to the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427040.png" /> by its coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427042.png" />, which are called absolute coordinates, or absolute components, of the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427043.png" />. The relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427044.png" /> of the geometric object relative to the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427045.png" /> are the absolute components of the geometric object into which the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427046.png" /> under consideration is converted by the transformation which changes the moving frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427047.png" /> into the absolute frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427048.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427051.png" /> are group parameters of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427052.png" />-parameter group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427053.png" />. The relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427054.png" /> of a fixed geometric object satisfy the completely-integrable system of differential equations

<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/g/g044/g044270/g04427055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427056.png" /> are left-invariant forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427058.png" />. The system (1) is called a system of differential equations of invariance of the geometric object, and also a system of differential equations of the representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427059.png" /> with invariant forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427060.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427061.png" /> are called the principal geometric object determining functions (or representation-determining functions). A system of the form (1) is a system of differential equations of invariance of a geometric object with relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427062.png" /> and group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427063.png" /> if and only if the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427064.png" /> are functions of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427065.png" /> alone and the given system (1) is completely integrable (the fundamental theorem in the theory of geometric objects). Fulfillment of the structure Lie equations for the object-determining functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427066.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/g/g044/g044270/g04427067.png" /></td> </tr></table>

are necessary and sufficient conditions for complete integrability of the system of differential equations of invariance of the geometric object. The differential forms

<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/g/g044/g044270/g04427068.png" /></td> </tr></table>

are called the structure forms of the representation, or the structure forms of the geometric object, with relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427069.png" />. The dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427070.png" /> of the representation space of a geometric object is taken to be the rank of the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427071.png" />. A necessary condition for a faithful transitive representation of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427072.png" />-parameter group on a space of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427073.png" /> is the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427074.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427076.png" /> is the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427077.png" />, is called the genre of the geometric object. The genre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427078.png" /> coincides with the number of independent absolute invariants of the geometric object.

The system of forms

<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/g/g044/g044270/g04427079.png" /></td> </tr></table>

where

<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/g/g044/g044270/g04427080.png" /></td> </tr></table>

is completely integrable. For a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427081.png" /> of a representation space of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427082.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/g/g044/g044270/g04427083.png" /></td> </tr></table>

and the resulting forms

<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/g/g044/g044270/g04427084.png" /></td> </tr></table>

satisfy the structure equations of a linear group.

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427085.png" /> of linearly independent forms among the forms

<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/g/g044/g044270/g04427086.png" /></td> </tr></table>

is an arithmetic invariant of the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427087.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427088.png" /> is called the character of isotropy of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427089.png" /> of the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427090.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427091.png" /> form a non-increasing sequence. There always exists a smallest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427092.png" /> such that

<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/g/g044/g044270/g04427093.png" /></td> </tr></table>

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427094.png" /> is also an arithmetic invariant of the representation space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427095.png" /> and is called the order of non-linearity of the geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427096.png" />.

If a system of differential equations of invariance

<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/g/g044/g044270/g04427097.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

contains a subsystem

<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/g/g044/g044270/g04427098.png" /></td> </tr></table>

then the system of components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g04427099.png" /> defines a geometric object — a subobject of the geometric object with relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270100.png" />.

If two geometric objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270102.png" /> are associated to the same group, and all relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270103.png" /> of one object can be represented by certain analytic functions in the relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270104.png" /> of the second object:

<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/g/g044/g044270/g044270105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

then one says that the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270106.png" /> is covered by the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270107.png" />. The geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270108.png" /> is called the covering geometric object, while the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270109.png" /> is said to be the covered geometric object. Two geometric objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270111.png" /> are called similar if each one covers the other. The ranks, genres, characteristics, and types of similar geometric objects are identical. A special case of similar geometric objects are isomers: geometric objects which only differ in the order of their components. If the system of differential equations of invariance of a geometric object is algebraically solvable with respect to all invariant forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270112.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270113.png" />, then any other object associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270114.png" /> can be covered by this object.

A system of functions (3) will be a system of relative components of a geometric object if and only if, in the system of differential equations which is satisfied by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270115.png" />, the coefficients of the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270116.png" /> in the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270117.png" /> are functions of these components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270118.png" /> only, i.e.

<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/g/g044/g044270/g044270119.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

If in the differential equations

<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/g/g044/g044270/g044270120.png" /></td> </tr></table>

which are satisfied by the components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270121.png" /> of a geometric object with relative components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270122.png" />, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270123.png" /> are homogeneous with respect to the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270124.png" />, then the system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270125.png" /> is said to be a truncated geometric object.

Let there be given a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270126.png" /> and an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270127.png" /> covered by it, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270128.png" />; then

<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/g/g044/g044270/g044270129.png" /></td> </tr></table>

The totality of relative components of the covering geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270130.png" />, of the covered geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270131.png" /> and the partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270132.png" /> of the latter with respect to the former form a system of relative components of a new covered geometric object:

<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/g/g044/g044270/g044270133.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

If equations (2) and (4) are valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270135.png" />, respectively, then

<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/g/g044/g044270/g044270136.png" /></td> </tr></table>

The geometric object (5) is called the derived geometric object.

A geometric object is called a linear or a quasi-tensorial object if the group of transformations of its components is linear, i.e.

<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/g/g044/g044270/g044270137.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270138.png" />, then the geometric object is called a linear homogeneous object, or a tensor. A geometric object is linear if and only if the principal functions which define it are of the form

<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/g/g044/g044270/g044270139.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270141.png" /> are constant. A geometric object is a linear homogeneous object if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270142.png" />, i.e.

<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/g/g044/g044270/g044270143.png" /></td> </tr></table>

A one-component tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270144.png" /> is called an invariant. The differential equation of an invariant has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270145.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270146.png" /> are constants. If not all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270147.png" /> are zero, the invariant is called relative. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270148.png" />, the invariant is called absolute.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270149.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270150.png" />-dimensional differentiable manifold and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270151.png" /> are the local coordinates of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270152.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270153.png" /> is some domain in this manifold, then it is always possible to introduce a completely-integrable system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270154.png" /> linear linearly independent differential forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270155.png" /> whose first integrals are the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270156.png" />. This means that

<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/g/g044/g044270/g044270157.png" /></td> </tr></table>

and

<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/g/g044/g044270/g044270158.png" /></td> </tr></table>

Consider a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270159.png" /> linear linearly independent forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270160.png" /> which satisfy the following structure equations:

<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/g/g044/g044270/g044270161.png" /></td> </tr></table>

<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/g/g044/g044270/g044270162.png" /></td> </tr></table>

The forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270163.png" /> have a fibre structure with respect to the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270164.png" /> and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270165.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270166.png" />, become invariant forms of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270167.png" />-parameter Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270168.png" /> with structure constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270169.png" />.

One says that a field of geometric objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270170.png" /> associated to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270171.png" /> (a field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270173.png" />-objects) is (locally) defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270174.png" /> if at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270175.png" /> of this manifold there is defined a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270176.png" /> associated to some Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270177.png" /> (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270178.png" />-object). Here

<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/g/g044/g044270/g044270179.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270180.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270181.png" />) are the absolute coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270182.png" />. Accordingly,

<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/g/g044/g044270/g044270183.png" /></td> </tr></table>

In other words, the system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270184.png" /> satisfying the system of differential equations

<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/g/g044/g044270/g044270185.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

is called a field of geometric objects associated to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270186.png" /> if the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270188.png" /> is completely integrable. The geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270189.png" /> is called the generating object of the field, and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270190.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270191.png" /> become the relative components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270192.png" />. The equations are called the differential equations of the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270193.png" />. The field of a geometric object determines a section in the associated fibre space, the base of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270194.png" />, and the fibres of which are the spaces of the given geometric object.

The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270195.png" /> are called the principal determining field functions, and the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270196.png" /> are called the complementary determining functions of the field of the geometric object (or the Pfaffian derivatives of the field). The totality of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270197.png" /> is also a system of relative components of the geometric object, which is called the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270198.png" /> or the extended geometric object of the first order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270199.png" />.

The system of differential equations (6) of the field of a geometric object is regularly prolongable along the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270200.png" />. This means that as a result of exterior differentiation of the system one obtains a quadratic system of the form

<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/g/g044/g044270/g044270201.png" /></td> </tr></table>

where

<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/g/g044/g044270/g044270202.png" /></td> </tr></table>

The system

<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/g/g044/g044270/g044270203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

together with (6) forms a system of differential equations of the field of the prolonged geometric object. The system (6)–(7) in turn is regularly prolongable. After the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270204.png" />-th prolongation one obtains a system of differential equations of the field of the extended geometric object of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270205.png" /> with relative components

<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/g/g044/g044270/g044270206.png" /></td> </tr></table>

If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270207.png" /> to which the geometric object is associated is the differential group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270208.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270209.png" />, the geometric object is called a differential-geometric object, while the field of such an object is called the field of the differential-geometric object.

If the generating object of a field covers another object (is covered by another object), the field of the former is called covering (covered), while the field of the latter is called covered (covering).

If a field of a geometric object has been defined on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270210.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270211.png" /> is said to be equipped, while the given field and the object generating it are called an equipping field and, correspondingly, an equipping object.

The field of an equipping geometric object induces a differential-geometric structure (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270212.png" />-structure in a wide sense of the word) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270213.png" />. For this reason an equipping object is also known as a structure object. The type of the structure is determined by the type of the structure object.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Whitehead,   "The foundations of differential geometry" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Laptev,   "Differential geometry of imbedded manifolds"  ''Trudy Moskov. Mat. Obshch.'' , '''2'''  (1953)  pp. 275–382  (In Russian)</TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270214.png" /> be a manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270215.png" /> a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270216.png" />. Quite generally, a geometric object at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270217.png" /> is a correspondence between (admissible) coordinate systems at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270218.png" /> and ordered sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270219.png" /> numbers (the components of the geometric object with respect to that coordinate system) such that

i) to each coordinate system at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270220.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270221.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270222.png" />, corresponds one set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270223.png" /> numbers;

ii) if the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270224.png" /> correspond to the coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270225.png" /> and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270226.png" /> correspond to a coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270227.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270228.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270229.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270230.png" /> give the local coordinate-change functions, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270231.png" /> are functions of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270232.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270233.png" />, and functionals of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270234.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270235.png" />, in an arbitrary small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270236.png" /> only.

If the coordinate transformations are all analytic, the requirement becomes that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270237.png" /> are functions of only the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270238.png" /> and the values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270239.png" /> and all their partial derivatives at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270240.png" />.

Thus, for example, tensors of various kinds (at a point) are geometric objects.

The functions linking the various sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270241.png" /> numbers are precisely the transformation rules of the geometric object in question. Geometric objects thus appear as the most general objects to be studied in differential geometry. Cf. [[#References|[a1]]], pp. 61ff for a great deal of material on geometric objects from this point of view.

Geometric objects as defined just above and geometric objects associated to a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270242.png" /> do not occur much in the modern mathematical literature. Instead one finds the related notions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270243.png" />-structures (cf. [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270244.png" />-structure]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270245.png" />-structures (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044270/g044270246.png" /> is a (Lie) pseudo-group, cf. [[Pseudo-group structure|Pseudo-group structure]]) and the topic of (local) transformation groups acting on manifolds [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.A. Schouten,   "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer  (1954)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Kobayashi,   "Transformation groups in differential geometry" , Springer  (1972)</TD></TR></table>