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 $ S $ of an $ r $- parameter Lie group $ G $ be put into correspondence with a transformation of each point $ M $ belonging to some domain $ D $ of a topological space $ E $, and let the zero element $ S _ {0} $ of the group correspond to the identity transformation (mapping) of the space into itself. Let a successive execution of transformations by two elements $ S _ {u} $ and $ S _ {v} $ 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 $ G $ is locally represented on $ E $ as a transformation group. The space $ E $ is called a representation space of the group $ G $ or a space with fundamental group $ G $.

A geometric object with given fundamental group $ G $, or a geometric object associated to a group $ G $( a $ G $- object for short), is defined as a point of the representation space of $ G $. 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 $ G $ are called equivalent if one of them may be transformed into the other by a transformation of $ G $. An intransitivity system is called a space of geometric objects in the proper sense. A representation space of $ G $ is called a homogeneous space with fundamental group $ G $ 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 $ X $. Let the representation space be a manifold and together with a frame $ R $ let it be subjected to all possible transformations of the fundamental group $ G $, then a complete family (space) of frames is obtained on which a simply-transitive representation of $ G $ is realized. This space is identified with the group space, or the parameter space, of $ G $. If an arbitrary point (frame) of this space is taken as reference point and is brought into correspondence with the unit element of $ G $, then all the points of this space are brought in a one-to-one correspondence with the elements of $ G $. 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 $ S _ {a} $ of the group is brought into correspondence with the frame $ R _ {a} $ obtained from an arbitrary fixed initial (absolute) frame $ R $ by right (left) shift by the element $ S _ {a} $: $ R _ {a} = RS _ {a} $. To a current element $ S _ {u} $ will correspond a current "moving" frame $ R _ {u} $. Each point $ X $ of the representation space of $ G $ is defined with respect to the frame $ R $ by its coordinates $ \widetilde{X} {} ^ {K} $, $ K = 1 \dots N $, which are called absolute coordinates, or absolute components, of the geometric object $ X $. The relative components $ X _ {u} ^ {K} $ of the geometric object relative to the frame $ R _ {u} = S _ {u} ^ {-} 1 R $ are the absolute components of the geometric object into which the object $ X $ under consideration is converted by the transformation which changes the moving frame $ R _ {u} $ into the absolute frame $ R $: $ X ^ {K} = S _ {u} \widetilde{X} {} ^ {K} $ and $ K ^ {K} = f ^ { K } ( u ^ {s} , \widetilde{X} {} ^ {J} ) $, where $ u ^ {s} $ are group parameters of the $ r $- parameter group, $ s = 1 \dots r $. The relative components $ X ^ {K} $ of a fixed geometric object satisfy the completely-integrable system of differential equations

$$ \tag{1 } dX ^ {K} - \xi _ {s} ^ {K} \omega ^ {s} = 0, $$

where $ \omega ^ {s} = \omega ^ {s} ( u, du) $ are left-invariant forms of $ G $ and $ \xi _ {s} ^ {K} = \xi _ {s} ^ {K} ( X ^ {J} ) $. 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 $ G $ with invariant forms $ \omega ^ {s} $. The functions $ \xi _ {s} ^ {K} $ 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 $ X ^ {K} $ and group $ G $ if and only if the coefficients $ \xi _ {s} ^ {K} $ are functions of the variables $ X ^ {K} $ 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 $ \xi _ {s} ^ {J} ( X ^ {K} ) $:

$$ \frac{\partial \xi _ {p} ^ {J} }{\partial X ^ {K} } \xi _ {q} ^ {K} - \frac{\partial \xi _ {q} ^ {J} }{\partial X ^ {K} } \xi _ {p} ^ {K} = \ C _ {pq} ^ {s} \xi _ {s} ^ {J} ,\ \ p, q, s = 1 \dots r, $$

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

$$ \Delta X ^ {J} \equiv \ dX ^ {J} - \xi _ {s} ^ {J} ( X) \omega ^ {s} $$

are called the structure forms of the representation, or the structure forms of the geometric object, with relative components $ X ^ {J} $. The dimension $ N $ of the representation space of a geometric object is taken to be the rank of the object $ X $. A necessary condition for a faithful transitive representation of an $ r $- parameter group on a space of objects $ X $ is the relation $ N \leq r $. The number $ \rho = N - R $, where $ R $ is the rank of the matrix $ ( \xi _ {s} ^ {K} ) $, is called the genre of the geometric object. The genre $ \rho $ coincides with the number of independent absolute invariants of the geometric object.

The system of forms

$$ \Delta X ^ {J} ,\ \Omega _ {K _ {1} } ^ {J} \dots \Omega _ {K _ {1} \dots K _ {a} } ^ {J} \dots $$

where

$$ \Omega _ {K _ {1} \dots K _ {a} } ^ {J} = \ \frac{\partial ^ {a} \xi _ {s} ^ {J} }{\partial X ^ {K _ {1} } \dots \partial X ^ {K _ {a} } } \omega ^ {s} ,\ \ a = 1, 2 \dots $$

is completely integrable. For a fixed point $ X _ {0} $ of a representation space of the group $ G $,

$$ \Delta X _ {0} ^ {J} \equiv \ - \xi _ {s} ^ {J} ( X _ {0} ) \omega ^ {s} = 0, $$

and the resulting forms

$$ \left . \overline \Omega \; {} _ {K _ {1} \dots K _ {a} } ^ {J} = \ \Omega _ {K _ {1} \dots K _ {a} } ^ {J} \right | _ {X ^ {J} = X _ {0} ^ {J} = \textrm{ const } } $$

satisfy the structure equations of a linear group.

The number $ r _ {m} $ of linearly independent forms among the forms

$$ \overline \Omega \; {} _ {K _ {1} } ^ {J} \dots \overline \Omega \; {} _ {K _ {1} \dots K _ {m} } ^ {J} $$

is an arithmetic invariant of the representation space of $ G $. The number $ \rho _ {m} = r - r _ {m} $ is called the character of isotropy of order $ m $ of the representation space of $ G $. The numbers $ \rho _ {a} $ form a non-increasing sequence. There always exists a smallest number $ q $ such that

$$ \rho _ {1} \geq \dots \geq \rho _ {q - 1 } \geq \rho _ {q} = \rho _ {q + 1 } = \dots . $$

The number $ q $ is also an arithmetic invariant of the representation space of $ G $ and is called the order of non-linearity of the geometric object $ X $.

If a system of differential equations of invariance

$$ \tag{2 } dX ^ {J} - \xi _ {s} ^ {J} ( X ^ {K} ) \omega ^ {s} = 0 $$

contains a subsystem

$$ dX ^ \alpha - \xi _ {s} ^ \alpha ( X ^ \beta ) \omega ^ {s} = 0,\ \ \alpha , \beta = 1 \dots n _ {1} < N, $$

then the system of components $ X ^ \alpha $ defines a geometric object — a subobject of the geometric object with relative components $ X ^ {J} $.

If two geometric objects $ X $ and $ Y $ are associated to the same group, and all relative components $ Y ^ \alpha $ of one object can be represented by certain analytic functions in the relative components $ X ^ {J} $ of the second object:

$$ \tag{3 } Y ^ \alpha = Y ^ \alpha ( X ^ {K} ), $$

then one says that the object $ Y $ is covered by the object $ X $. The geometric object $ X $ is called the covering geometric object, while the object $ Y $ is said to be the covered geometric object. Two geometric objects $ X _ {1} $ and $ X _ {2} $ 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 $ \omega ^ {s} $ of the group $ G $, then any other object associated to $ G $ 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 $ Y ^ \alpha $, the coefficients of the decomposition of $ dY ^ \alpha $ in the forms $ \omega ^ {s} $ are functions of these components $ Y ^ \alpha $ only, i.e.

$$ \tag{4 } dY ^ \alpha - \eta _ {s} ^ \alpha ( Y ^ \beta ) \omega ^ {s} = 0. $$

If in the differential equations

$$ dX ^ {a} - \xi _ {s} ^ {a} ( X ^ {J} ) \omega ^ {s} = 0,\ \ a = 1 \dots n _ {2} < N, $$

which are satisfied by the components $ X ^ {a} $ of a geometric object with relative components $ X ^ {J} $, the functions $ \xi _ {s} ^ {a} ( X ^ {J} ) $ are homogeneous with respect to the components of $ X ^ {a} $, then the system of functions $ X ^ {a} $ is said to be a truncated geometric object.

Let there be given a geometric object $ X $ and an object $ Y $ covered by it, i.e. $ Y ^ \alpha = Y ^ \alpha ( X ^ {J} ) $; then

$$ Y _ {K} ^ \alpha \equiv \ \frac{\partial Y ^ \alpha }{\partial X ^ {K} } = \ F _ {K} ^ { \alpha } ( X ^ {J} ). $$

The totality of relative components of the covering geometric object $ X ^ {J} $, of the covered geometric object $ Y ^ \alpha $ and the partial derivatives $ Y _ {K} ^ \alpha $ of the latter with respect to the former form a system of relative components of a new covered geometric object:

$$ \tag{5 } \left \{ X ^ {J} , Y ^ \alpha ,\ \frac{\partial Y ^ \alpha }{\partial X ^ {K} } \right \} . $$

If equations (2) and (4) are valid for $ X ^ {J} $ and $ Y ^ \alpha $, respectively, then

$$ dY _ {K} ^ \alpha - \left ( \frac{\partial \eta _ {s} ^ \alpha }{\partial Y ^ \beta } Y _ {K} ^ \beta - \frac{\partial \xi _ {s} ^ {J} }{\partial X ^ {K} } Y _ {J} ^ \alpha \right ) \omega ^ {s} = 0. $$

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.

$$ \widetilde{X} {} ^ {J} = \ B _ {K} ^ {J} ( u) X ^ {K} + B ^ {J} ( u). $$

If $ B ^ {J} = 0 $, 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

$$ \xi _ {s} ^ {J} = \ K _ {sK} ^ {J} X ^ {K} + K _ {s} ^ {J} , $$

where $ K _ {sK} ^ {J} $, $ K _ {s} ^ {J} $ are constant. A geometric object is a linear homogeneous object if and only if $ K _ {s} ^ {J} = 0 $, i.e.

$$ \xi _ {s} ^ {J} = \ K _ {sK} ^ {J} X ^ {K} . $$

A one-component tensor $ X $ is called an invariant. The differential equation of an invariant has the form $ dX - XK _ {s} \omega ^ {s} = 0 $, where $ K _ {s} $ are constants. If not all $ K _ {s} $ are zero, the invariant is called relative. If $ K _ {s} = 0 $, the invariant is called absolute.

If $ V _ {n} $ is an $ n $- dimensional differentiable manifold and if $ u _ {i} $ are the local coordinates of a point $ u \in U \subset V _ {n} $, where $ U $ is some domain in this manifold, then it is always possible to introduce a completely-integrable system of $ n $ linear linearly independent differential forms $ \theta ^ {i} $ whose first integrals are the coordinates $ u ^ {i} $. This means that

$$ \theta ^ {i} = \ u _ {k} ^ {i} du ^ {k} $$

and

$$ D \theta ^ {i} = \ \theta ^ {k} \wedge \theta _ {k} ^ {i} . $$

Consider a system of $ r $ linear linearly independent forms $ \omega ^ \alpha $ which satisfy the following structure equations:

$$ D \omega ^ \alpha = \ { \frac{1}{2} } C _ {\beta \gamma } ^ \alpha ( u) \omega ^ \beta \wedge \omega ^ \gamma + \theta ^ {k} \wedge \omega _ {k} ^ \alpha , $$

$$ \alpha , \beta , . . . = n + 1 \dots n + r. $$

The forms $ \omega ^ \alpha $ have a fibre structure with respect to the forms $ \theta ^ {i} $ and, if $ u ^ {i} = u _ {0} ^ {i} $, i.e. if $ \theta ^ {k} = 0 $, become invariant forms of an $ r $- parameter Lie group $ G $ with structure constants $ C _ {\beta \gamma } ^ \alpha ( u _ {0} ^ {i} ) $.

One says that a field of geometric objects $ \{ X \} $ associated to the group $ G $( a field of $ G $- objects) is (locally) defined on $ V _ {n} $ if at any point $ u \in U $ of this manifold there is defined a geometric object $ \{ X \} $ associated to some Lie group $ G $( a $ G $- object). Here

$$ \widetilde \Phi {} ^ {J} = \ \widetilde \Phi {} ^ {J} ( u ^ {1} \dots u ^ {n} ), $$

where $ \widetilde \Phi {} ^ {J} $( $ J = 1 \dots N $) are the absolute coordinates of $ \{ X \} $. Accordingly,

$$ d \widetilde \Phi = \ \widetilde \Phi {} _ {k} ^ {J} \theta ^ {k} . $$

In other words, the system of functions $ X ^ {J} $ satisfying the system of differential equations

$$ \tag{6 } \Delta X ^ {J} \equiv \ dX ^ {J} - \Xi _ \alpha ^ {J} ( X ^ {K} ) \omega ^ \alpha - X _ {k} ^ {J} \theta ^ {k} = 0 $$

is called a field of geometric objects associated to the group $ G $ if the system $ \Delta X ^ {J} = 0 $, $ \theta ^ {k} = 0 $ is completely integrable. The geometric object $ \{ X \} $ is called the generating object of the field, and the functions $ X ^ {J} $ at $ \theta ^ {k} = 0 $ become the relative components of $ \{ X \} $. The equations are called the differential equations of the field of $ \{ X \} $. The field of a geometric object determines a section in the associated fibre space, the base of which is $ V _ {n} $, and the fibres of which are the spaces of the given geometric object.

The functions $ \Xi _ {s} ^ {J} ( X ^ {K} ) $ are called the principal determining field functions, and the coefficients $ X _ {k} ^ {J} $ are called the complementary determining functions of the field of the geometric object (or the Pfaffian derivatives of the field). The totality of functions $ X ^ {J} , X _ {k} ^ {J} $ is also a system of relative components of the geometric object, which is called the extension of $ \{ X \} $ or the extended geometric object of the first order of $ \{ X \} $.

The system of differential equations (6) of the field of a geometric object is regularly prolongable along the forms $ \theta ^ {k} $. This means that as a result of exterior differentiation of the system one obtains a quadratic system of the form

$$ \Delta X _ {i} ^ {J} \wedge \theta ^ {i} = 0, $$

where

$$ \Delta X _ {i} ^ {J} \equiv \ dX _ {i} ^ {J} - \Xi _ {i \alpha } ^ {J} ( X ^ {K} , X _ {k} ^ {K} ) \omega ^ \alpha . $$

The system

$$ \tag{7 } \Delta X _ {i} ^ {J} = \ X _ {ik} ^ {J} \theta ^ {k} $$

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 $ q $- th prolongation one obtains a system of differential equations of the field of the extended geometric object of order $ q $ with relative components

$$ \{ X ^ {J} , X _ {k} ^ {J} \dots X _ {k _ {1} \dots k _ {q} } ^ {J} \} . $$

If the group $ G $ to which the geometric object is associated is the differential group $ \mathop{\rm GL} ^ {p} ( n, R) $ of order $ p $, 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 $ V _ {n} $, then $ V _ {n} $ 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 $ G $- structure in a wide sense of the word) on $ V _ {n} $. 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.

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Let $ M $ be a manifold, $ m \in M $ a point of $ M $. Quite generally, a geometric object at $ m \in M $ is a correspondence between (admissible) coordinate systems at $ m \in M $ and ordered sets of $ N $ numbers (the components of the geometric object with respect to that coordinate system) such that

i) to each coordinate system at $ m $, $ \phi : U \rightarrow \mathbf R ^ {n} $, $ m \in M $, corresponds one set of $ N $ numbers;

ii) if the numbers $ g _ {1} \dots g _ {N} $ correspond to the coordinate system $ \phi $ and the numbers $ g _ {1} ^ \prime \dots g _ {N} ^ \prime $ correspond to a coordinate system $ \phi ^ \prime : U ^ \prime \rightarrow \mathbf R ^ {n} $, $ m \in U ^ \prime $, and $ f _ {i} = \phi _ {i} ^ \prime \circ \phi _ {i} ^ {-} 1 $ on $ \phi _ {i} ( U \cap U ^ \prime ) $ give the local coordinate-change functions, then the $ g _ {j} ^ \prime $ are functions of the $ g _ {j} $, $ j = 1 \dots N $, and functionals of the $ f _ {i} $, $ i = 1 \dots n $, in an arbitrary small neighbourhood of $ m $ only.

If the coordinate transformations are all analytic, the requirement becomes that the $ g _ {j} ^ \prime $ are functions of only the $ g _ {j} $ and the values of the $ f _ {i} $ and all their partial derivatives at $ \phi ( m) $.

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

The functions linking the various sets of $ N $ 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. [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 $ G $ do not occur much in the modern mathematical literature. Instead one finds the related notions of $ G $- structures (cf. $ G $- structure), $ \Gamma $- structures (where $ \Gamma $ is a (Lie) pseudo-group, cf. Pseudo-group structure) and the topic of (local) transformation groups acting on manifolds [a2].

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