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

$$ f: X \rightarrow Y,\ \ X, Y \in K ( G), $$

such that for any $ g \in G $ the following diagram is commutative:

$$ \begin{array}{rcl} X &\rightarrow ^ { f } & Y \\ {l _ {g} } \downarrow &{} &\downarrow {l _ {g} ^ {1} } \\ X &\rightarrow _ { f } & Y \\ \end{array} $$

Here $ l _ {g} $ and $ l _ {g} ^ {1} $ are the transformations of the $ G $-spaces $ X $ and $ Y $, respectively, determined by $ g $. In this case one says that $ Y $ is a restriction of $ X $ by means of $ f $, or that $ X $ is a prolongation of $ Y $. The class $ K ( G) $ becomes a category with the $ G $-restrictions as morphisms.

Examples of $ G $-restrictions.

1) Let $ T ( p, q) \in K ( \mathop{\rm GL} ( n, \mathbf R )) $ be the space of tensors of type $ ( p, q) $, $ p, q \geq 1 $. The contraction mapping

$$ T ( p, q) \rightarrow T ( p - 1, q - 1) $$

is a restriction. The complete contraction of tensors of $ T ( p, p) $,

$$ T ( p, p): T ( p, p) \rightarrow \mathbf R , $$

is an example of a restriction invariant.

2) If $ X, Y \in K ( G) $, then $ X \times Y $ restricts by means of $ \mathop{\rm pr} _ {X} $ and $ \mathop{\rm pr} _ {Y} $, respectively, to $ X $ and $ Y $. In other words, $ X \times Y $ is a prolongation of both $ X $ and $ Y $.

The concept of a restriction can be naturally generalized to classes of fibre spaces associated with principal fibre bundles. Let $ \pi : P ( M, H) \rightarrow M $ be a principal fibre bundle with structure group $ H $, acting on $ P $ from the right, and let $ F \in K ( H) $ be a left $ H $-space. Fibre spaces associated with $ P $ by objects from $ K ( P) $ are spaces of the type

$$ F ( P) = ( P \times F )/H, $$

where factorization is by the following right action of $ H $ on $ P \times F $:

$$ ( \xi , Y) h = ( \xi h, h ^ {-1} Y),\ \ ( \xi , Y) \in P \times F,\ \ h \in H. $$

The space $ F ( P) \in K ( P) $ is a fibre bundle over the base $ M $ with typical fibre $ F $. The element $ y \in F ( P) $ determined by a pair $ ( \xi , Y) \in P \times F $ is written as $ y = \xi Y $. If $ F, \Phi \in K ( H) $ and $ f: F \rightarrow \Phi $ is an $ H $-restriction mapping, then, by construction, $ F ( P) $ and $ \Phi ( P) f $ induce a fibre-wise surjective mapping $ f: F ( P) \rightarrow \Phi ( P) $, called a $ P $-restriction. The $ P $-restriction $ \widetilde{f} $ is defined by

$$ \widetilde{f} ( \xi Y) = \xi f ( Y),\ \ \xi \in P,\ Y \in F. $$

Thus, the class $ K ( P) $ of fibre bundles associated with $ P $ is a category with $ P $-restrictions $ \widetilde{f} $ as morphisms. The correspondence $ F \mapsto F ( P) $, $ f \mapsto \widetilde{f} $ is a bijective functor from the category $ K ( H) $ to the category $ K ( P) $. Hence it is sufficient to study the restriction operation in the category of $ H $-spaces.

If $ s: M \rightarrow F ( P) $ is a section of a fibre bundle $ F ( P) $ (a field of geometric objects of type $ F $), then the $ P $-restriction $ \widetilde{f} : F ( P) \rightarrow \Phi ( P) $ associates the section $ \widetilde{s} = \widetilde{f} \circ s $ of the restricted fibre bundle $ \Phi ( P) $ to $ s $. In other words, the field of geometric objects $ s ( x) $, $ x \in M $, restricts the field of geometric objects $ \widetilde{f} \circ s ( x) $. If $ s ( x) $ is the structure object of a $ G $-structure, then the study of the $ G $-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

References

[a1]	G. Jensen, "Higher order contact of submanifolds of homogeneous spaces" , Lect. notes in math. , 610 , Springer (1977)
[a2]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1979)