Method of extensions and restrictions

method of prolongations and restrictions

A method for studying various differential-geometric structures (cf. Differential-geometric structure) on smooth manifolds and their submanifolds. At the basis of this method there lies a differential-algebraic criterion for an operation that allows one to associate in an invariant (coordinate-free) way to a given structure structures intrinsically related to it, among them their differential invariants (cf. Differential invariant). Historically this method arose as a consequence of the moving-frame method as an invariant method for studying submanifolds of homogeneous spaces or of spaces with a connection. Subsequently the method of prolongations and restrictions was extended to the geometry of arbitrary fibre spaces (cf. Fibre space). In distinction from the aim of the moving-frame method — to construct a canonical field of frames and differential invariants of the unknown structure by means of subsequent restriction of corresponding principal fibre spaces — the method of prolongations and restrictions has as its aim the construction of invariants and invariantly associated structures without restricting the principal fibres of frames. The process of canonization of a frame is included in the method of prolongations and restrictions.

Let be a Lie group and let be the class of -spaces with a left action of as transformation group on them. A -restriction is a smooth surjective mapping

such that for any the following diagram is commutative:

Here and are the transformations of the -spaces and , respectively, determined by . In this case one says that is a restriction of by means of , or that is a prolongation of . The class becomes a category with the -restrictions as morphisms.

Examples of -restrictions.

1) Let be the space of tensors of type , . The contraction mapping

is a restriction. The complete contraction of tensors of ,

is an example of a restriction invariant.

2) If , then restricts by means of and , respectively, to and . In other words, is a prolongation of both and .

The concept of a restriction can be naturally generalized to classes of fibre spaces associated with principal fibre bundles. Let be a principal fibre bundle with structure group , acting on from the right, and let be a left -space. Fibre spaces associated with by objects from are spaces of the type

where factorization is by the following right action of on :

The space is a fibre bundle over the base with typical fibre . The element determined by a pair is written as . If and is an -restriction mapping, then, by construction, and induce a fibre-wise surjective mapping , called a -restriction. The -restriction is defined by

Thus, the class of fibre bundles associated with is a category with -restrictions as morphisms. The correspondence , is a bijective functor from the category to the category . Hence it is sufficient to study the restriction operation in the category of -spaces.

If is a section of a fibre bundle (a field of geometric objects of type ), then the -restriction associates the section of the restricted fibre bundle to . In other words, the field of geometric objects , , restricts the field of geometric objects . If is the structure object of a -structure, then the study of the -structures and its invariants reduces largely to the search for restricting geometric objects. In the latter process, an important role is played by differential criteria for restrictions, formulated in terms of structure differential forms of fibre spaces forming the base of the method of restrictions and prolongations.

References

 [1] G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential-geometric investigation" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian) [2] G.F. Laptev, , Proc. 3-rd All-Union Mat. Congress (Moscow, 1956) , 3 , Moscow (1958) pp. 409–418