Structure(2)

Also called mathematical structure. A generic name for unifying concepts whose general characteristic is that they can be applied to sets whose elements are of an indefinite nature. In order to define a structure, relations are given in which the elements of the set appear (the type characteristic of a structure), and it is then postulated that these relations satisfy certain conditions — axioms of the structure.

References

[1]	N. Bourbaki, "Eléments d'histoire des mathématiques" , Hermann (1960)
[2]	N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)

Comments

Sets endowed with a given structure plus mappings of sets which preserve this structure together form a category. Such categories are called concrete (cf. also Category; Sets, category of). More precisely, a concrete category is a pair $ ( {\mathcal C}, U) $ consisting of a category $ {\mathcal C} $ and a faithful functor $ U: {\mathcal C} \rightarrow \mathop{\rm Set} $. Because $ U $ is faithful, $ f $ can be identified with $ Uf $, and an object $ C $ of a concrete category is a set $ U( C) $ with extra structure while a morphism $ f $ is an actual mapping of sets that preserves the extra structure; composition of morphisms is accomplished by the usual composition of mappings of sets. Often the set of morphisms $ {\mathcal C} ( C, D) $ consists of all structure-preserving mappings of sets, but this need not be the case.

A category is concrete if and only if it satisfies the Isbell condition (the Freyd concreteness theorem). Here, the Isbell condition is the following. A span in a category is a diagram of the form

$$ \begin{array}{lcr} {} & C &{} \\ {} _ {f} \swarrow &{} &\searrow _ {g} \\ A &{} & B \\ \end{array} $$

Two $ ( A, B) $- spans $ ( f, g) $ and $ ( f ^ { \prime } , g ^ \prime ) $ are equivalent if for all pairs of morphisms $ ( p: A \rightarrow D, q: B \rightarrow D) $ either both diagrams

$$ \begin{array}{lcr} {} & C &{} \\ {} _ {f} \swarrow &{} &\searrow _ {g} \\ A &{} & B \\ {} _ {p} \searrow &{} &\swarrow _ {q} \\ {} & D &{} \\ \end{array} \ \ \begin{array}{lcr} {} & C &{} \\ {} _ {f ^ { \prime } } \swarrow &{} &\searrow _ {g ^ \prime } \\ A &{} & B \\ \searrow _ {p} &{} &\swarrow _ {q} \\ {} & D &{} \\ \end{array} $$

commute or both do not commute. A category satisfies the Isbell condition if for all objects $ ( A, B) $ there exists a set of $ ( A, B) $- spans $ M _ {A,B } $ such that each $ ( A, B) $- span is equivalent to precisely one element of $ M _ {A,B } $.

A structure is also an obsolete term for lattice.

References

[a1]	J. Adamek, "Theory of mathematical structures" , Reidel (1983) pp. Chapt. 6
[a2]	S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. 26
[a1]	A.G. Kurosh, "Theory of groups" , 2 , Chelsea, reprint (1955) pp. 85 (Translated from Russian)

A structure on a manifold, a geometric quantity, a geometric object, or a field of geometric objects, is a section of a bundle associated with the principal bundle of coframes on the manifold $ M $. Intuitively, a geometric quantity can be considered as a quantity whose value depends not only on the point $ x $ of the manifold $ M $, but also on the choice of the coframe — an infinitesimal system of coordinates at the point $ x $( see Chart).

More precisely, let $ \mathop{\rm GL} ^ {k} ( n) $ be the general differential group of order $ k $( the group of $ k $- jets at zero of transformations of the space $ \mathbf R ^ {n} $ that preserve the origin), and let $ M _ {k} $ be the manifold of coframes of order $ k $ of an $ n $- dimensional manifold $ M $( i.e. the manifold of $ k $- jets $ j _ {x} ^ {k} ( u) $ of local charts $ u: M \supset U \rightarrow \mathbf R ^ {n} $ with origin at the point $ x = u ^ {-} 1 ( 0) $). The group $ \mathop{\rm GL} ^ {k} ( u) $ acts from the left on $ M _ {k} $ by

$$ j _ {0} ^ {k} ( \phi ) j _ {0} ^ {k} ( u) = \ j _ {x} ^ {k} ( \phi \circ u),\ \ j _ {0} ^ {k} ( \phi ) \in \mathop{\rm GL} ^ {k} ( n),\ \ j _ {x} ^ {k} ( u) \in M _ {k} , $$

and this action defines on $ M _ {k} $ the structure of a principal $ \mathop{\rm GL} ^ {k} ( n) $- bundle $ \pi _ {k} : M _ {k} \rightarrow M $, which is called the bundle of coframes of order $ k $. Let $ W $ be an arbitrary $ \mathop{\rm GL} ^ {k} ( n) $- manifold, i.e. a manifold with a left action of the group $ \mathop{\rm GL} ^ {k} ( n) $. Finally, let $ W( M) $ be the orbit space of the left action of $ \mathop{\rm GL} ^ {k} ( n) $ on $ M _ {k} \times W $, while $ \pi _ {W} $ is its natural projection onto $ M $. The bundle $ \pi _ {W} : W( M) \rightarrow M $( associated with $ M _ {k} $ and $ W $) is called a bundle of geometric structures of order $ \leq k $ and of type $ W $, while its sections are called structures of type $ W $. Structures of type $ W $ are in a natural one-to-one correspondence with $ \mathop{\rm GL} ^ {k} ( n) $- equivariant mappings $ S: M _ {k} \rightarrow W $. Thus, a structure of type $ W $ can be seen as a $ W $- valued function $ S $ on the manifold $ M _ {k} $ of $ k $- frames that satisfies the following condition of equivariance:

$$ S( gu ^ {k} ) = gS( u ^ {k} ),\ \ g \in \mathop{\rm GL} ^ {k} ( n),\ \ u ^ {k} \in M _ {k} . $$

The bundle $ \pi _ {W} $ of geometric objects is a natural bundle in the sense that the diffeomorphism group of $ M $ acts as the automorphism group of $ \pi _ {W} $.

If $ W $ is a vector space with a linear (or affine) action of $ \mathop{\rm GL} ^ {k} ( n) $, then a structure of type $ W $ is said to be linear (or affine).

A basic example of a linear structure of order one is a tensor structure, or a tensor field. Let $ V = \mathbf R ^ {n} $, $ V ^ \star = \mathop{\rm Hom} ( V, \mathbf R ) $ and let $ V _ {q} ^ {p} = ((\otimes ^ {p} V)) \otimes ((\otimes ^ {q} V ^ \star )) $ be the space of tensors of type $ ( p, q) $ with the natural tensor representation of $ \mathop{\rm GL} ^ {1} ( n) = \mathop{\rm GL} ( n) $. A structure of type $ V _ {q} ^ {p} $ is called a tensor field of type $ ( p, q) $. It can be regarded as a vector function on the manifold of coframes $ M _ {1} $ which assigns to the coframe $ \theta = j _ {x} ^ {1} ( u) = ( du ^ {1} \dots du ^ {n} ) $ the set of coordinates $ S( \theta ) _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $ of the tensor $ S( \theta ) \in V _ {q} ^ {p} $, relative to the standard basis

$$ \{ e _ {i _ {1} } \otimes \dots \otimes e _ {i _ {p} } \otimes e ^ {\star j _ {1} } \otimes {} \dots \otimes e ^ {\star j _ {q} } \} $$

of $ V _ {q} ^ {p} $. Given a linear transformation of coframes $ \theta \rightarrow g \theta = ( g _ {a} ^ {i} du ^ {a} ) $, the coordinates $ S _ {j _ {1} {} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } $ are transformed in accordance with the tensor representation:

$$ S _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } ( g \theta ) = \ g _ {a _ {1} } ^ {i _ {1} } \dots g _ {a _ {p} } ^ {i _ {p} } ( g ^ {-} 1 ) _ {j _ {1} } ^ {b _ {1} } \dots ( g ^ {-} 1 ) _ {j _ {q} } ^ {b _ {q} } S ( \theta ) _ {b _ {1} \dots b _ {q} } ^ {a _ {1} \dots a _ {p} } . $$

The most important examples of tensor structures are a vector field, a Riemannian metric, a differential form, a symplectic structure, a complex structure, and most commonly, an affinor. All linear structures (of whatever order) are exhausted by Rashevskii super-tensors (see [4]). An example of an affine structure of order two is an affine connection without torsion, which can be regarded as a structure of type $ V _ {(} 2) ^ {1} $, where $ V _ {(} 2) ^ {1} \approx V \otimes S ^ {2} V ^ \star $ is the kernel of the natural homomorphism $ \mathop{\rm GL} ^ {2} ( n) \rightarrow \mathop{\rm GL} ^ {1} ( n) $, considered as a vector space with the natural action of $ \mathop{\rm GL} ^ {2} ( n) = \mathop{\rm GL} ( n) V _ {(} 2) ^ {1} $. A large and important class of structures is the class of infinitesimally-homogeneous structures or $ G $- structures (cf. $ G $- structure) — structures of type $ W $, where $ W = \mathop{\rm GL} ^ {k} ( n)/G $ is a homogeneous space of the group $ \mathop{\rm GL} ^ {k} ( n) $.

The above definition of a structure is not sufficiently general, and does not include a number of important geometric structures such as a spinor structure, a symplectic spinor structure, etc. A natural generalization is to study generalized $ G $- structures that are principal bundles with a fixed homomorphism onto a $ G $- structure, and sections of associated bundles.

References

[1]	P. Rashevskii, "Caractères tensoriels de l'espace sousprojectif" Trudy Sem. Vektor. i Tenzor. Anal. , 1 (1933) pp. 126–142
[2]	V. Vagner, "The theory of geometric objects and the theory of finite and infinite continuous transformation groups" Dokl. Akad. Nauk SSSR , 46 : 9 (1945) pp. 347–349 (In Russian)
[3]	O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932)
[4]	P.K. Rashevskii, "On linear representations of differential groups and Lie groups with nilpotent radical" Trudy Moskov. Mat. Obshch. , 6 (1957) pp. 337–370 (In Russian)
[5]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[6]	Ch. Ehresmann, "Introduction à la théorie des structures infinitésimales et des pseudo-groupes de Lie" , Géométrie Diff. Coll. Internat. C.N.R.S. (1953) pp. 97–110

D.V. Alekseevskii

Comments

Historically, E. Cartan was the first to introduce the concept of a structure.

References

[a1]	E. Cartan, "La théorie des groupes et les recherches récentes de géométrie différentielle" Enseign. Math. , 24 (1925) pp. 5–18