In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the higher order frame bundle F r ( M ) {\displaystyle F^{r}(M)}, for some r ≥ 1 {\displaystyle r\geq 1}. In other words, its transition functions depend functionally on local changes of coordinates in the base manifold M {\displaystyle M} together with their partial derivatives up to order at most r {\displaystyle r}.

The concept of a natural bundle was introduced in 1972 by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.

Definition

Let M f {\displaystyle {\mathcal {M}}f} denote the category of smooth manifolds and smooth maps and M f n {\displaystyle {\mathcal {M}}f_{n}} the category of smooth n {\displaystyle n}-dimensional manifolds and local diffeomorphisms. Consider also the category F M {\displaystyle {\mathcal {FM}}} of fibred manifolds and bundle morphisms, and the functor B : F M → M f {\displaystyle B:{\mathcal {FM}}\to {\mathcal {M}}f} associating to any fibred manifold its base manifold.

A natural bundle (or bundle functor) is a functor F : M f n → F M {\displaystyle F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}} satisfying the following three properties:

  1. B ∘ F = i d {\displaystyle B\circ F=\mathrm {id} }, i.e. F ( M ) {\displaystyle F(M)} is a fibred manifold over M {\displaystyle M}, with projection denoted by p M : F ( M ) → M {\displaystyle p_{M}:F(M)\to M};
  2. if U ⊆ M {\displaystyle U\subseteq M} is an open submanifold, with inclusion map i : U ↪ M {\displaystyle i:U\hookrightarrow M}, then F ( U ) {\displaystyle F(U)} coincides with p M − 1 ( U ) ⊆ F ( M ) {\displaystyle p_{M}^{-1}(U)\subseteq F(M)}, and F ( i ) : F ( U ) → F ( M ) {\displaystyle F(i):F(U)\to F(M)} is the inclusion p − 1 ( U ) ↪ F ( M ) {\displaystyle p^{-1}(U)\hookrightarrow F(M)};
  3. for any smooth map f : P × M → N {\displaystyle f:P\times M\to N} such that f ( p , ⋅ ) : M → N {\displaystyle f(p,\cdot ):M\to N} is a local diffeomorphism for every p ∈ P {\displaystyle p\in P}, then the function P × F ( M ) → F ( N ) , ( p , x ) ↦ F ( f ( p , ⋅ ) ) ( x ) {\displaystyle P\times F(M)\to F(N),(p,x)\mapsto F(f(p,\cdot ))(x)} is smooth.

As a consequence of the first condition, one has a natural transformation p : F → i d M f n {\displaystyle p:F\to \mathrm {id} _{{\mathcal {M}}f_{n}}}.

Finite order natural bundles

A natural bundle F : M f n → F M {\displaystyle F:{\mathcal {M}}f_{n}\to {\mathcal {FM}}} is called of finite order r {\displaystyle r} if, for every local diffeomorphism f : M → N {\displaystyle f:M\to N} and every point x ∈ M {\displaystyle x\in M}, the map F ( f ) x : F ( M ) x → F ( N ) f ( x ) {\displaystyle F(f)_{x}:F(M)_{x}\to F(N)_{f(x)}} depends only on the jet j x r f {\displaystyle j_{x}^{r}f}. Equivalently, for every local diffeomorphisms f , g : M → N {\displaystyle f,g:M\to N} and every point x ∈ M {\displaystyle x\in M}, one hasj x r f = j x r g ⇒ F ( f ) | F ( M ) x = F ( g ) | F ( M ) x . {\displaystyle j_{x}^{r}f=j_{x}^{r}g\Rightarrow F(f)|_{F(M)_{x}}=F(g)|_{F(M)_{x}}.}Natural bundles of order r {\displaystyle r} coincide with the associated fibre bundles to the r {\displaystyle r}-th order frame bundles F r ( M ) {\displaystyle F^{r}(M)}.

After various intermediate cases, it was proved by Epstein and Thurston that all natural bundles have finite order.

Natural Γ {\displaystyle \Gamma } -bundles

The notion of natural Γ {\displaystyle \Gamma }-bundle arises from that of natural bundle by restricting to the suitable categories of Γ {\displaystyle \Gamma }-manifolds and of Γ {\displaystyle \Gamma }-fibred manifolds, where Γ {\displaystyle \Gamma } is a pseudogroup. The case when Γ {\displaystyle \Gamma } is the pseudogroup of all diffeomorphisms between open subsets of R n {\displaystyle \mathbb {R} ^{n}} recovers the ordinary notion of natural bundle.

Under suitable assumptions, natural Γ {\displaystyle \Gamma }-bundles have finite order as well.

Examples

An example of natural bundle (of first order) is the tangent bundle T M {\displaystyle TM} of a manifold M {\displaystyle M}.

Other examples include the cotangent bundles, the bundles of metrics of signature ( r , s ) {\displaystyle (r,s)} and the bundle of linear connections.

Notes

  • Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), (PDF), Springer-Verlag, archived from (PDF) on 2017-03-30
  • Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
  • Saunders, D.J. (1989), , Cambridge University Press, ISBN 0-521-36948-7