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Natural bundle

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In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle F s ( M ) {\displaystyle F^{s}(M)} for some s ≥ 1 {\displaystyle s\geq 1} . It turns out that 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 s {\displaystyle s} .

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

Definition

Let M f {\displaystyle Mf} denote the category of smooth manifolds and smooth maps and M f n {\displaystyle Mf_{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. B ( M ) {\displaystyle B(M)} is a fibred manifold over M {\displaystyle M} , with projection denoted by p M : B ( M ) → M {\displaystyle p_{M}:B(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 → B {\displaystyle p:F\to B} .

Finite order natural bundles

A natural bundle F : M f n → M f {\displaystyle F:Mf_{n}\to Mf} 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 has j 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 s ( M ) {\displaystyle F^{s}(M)} .

A classical result by Epstein and Thurston shows that all natural bundles have finite order.

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

  1. Palais, Richard; Terng, Chuu-Lian (1977), "Natural bundles have finite order", Topology, 16: 271–277, doi:10.1016/0040-9383(77)90008-8, hdl:10338.dmlcz/102222
  2. A. Nijenhuis (1972), Natural bundles and their general properties, Tokyo: Diff. Geom. in Honour of K. Yano, pp. 317–334
  3. Epstein, D. B. A.; Thurston, W. P. (1979). "Transformation Groups and Natural Bundles". Proceedings of the London Mathematical Society. s3-38 (2): 219–236. doi:10.1112/plms/s3-38.2.219.
  4. Fatibene, Lorenzo; Francaviglia, Mauro (2003). Natural and Gauge Natural Formalism for Classical Field Theorie. Springer. doi:10.1007/978-94-017-2384-8. ISBN 978-1-4020-1703-2.

References

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