Natural bundle explained

In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle

F ^{s(M)} for some s \geq 1 . It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold M together with their partial derivatives up to order at most 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

Mf denote the category of smooth manifolds and smooth maps and Mf _{n} the category of smooth n -dimensional manifolds and local diffeomorphisms. Consider also the category l{FM} of fibred manifolds and bundle morphisms, and the functor B: l{FM} \to l{M}f associating to any fibred manifold its base manifold.F: l{M}f _{n} \to l{FM} satisfying the following three properties:B \circ F = id , i.e. B(M) is a fibred manifold over M , with projection denoted by p _{M:} B(M) \to M ;

1. if

U \subseteq M is an open submanifold, with inclusion map i: U \hookrightarrow M , then F(U) coincides with p ^{-1} _{M} (U) \subseteq F(M) , and F(i): F(U) \to F(M) is the inclusion p ^{-1} (U) \hookrightarrow F(M) ;

1. for any smooth map

f: P x M \to N such that f (p, ⋅ ): M \to N is a local diffeomorphism for every p \in P , then the function P x F(M) \to F(N), (p,x) \mapsto F(f (p, ⋅ )) (x) is smooth.p: F \to B .

Finite order natural bundles

A natural bundle

F: Mf _{n} \to Mf is called of finite order r if, for every local diffeomorphism f: M \to N and every point x \in M , the map F(f) _{x:} F(M) _{x} \to F(N) _{f(x)} depends only on the jet j ^{r} _{x} f . Equivalently, for every local diffeomorphisms f,g: M \to N and every point x \in M , one has$$j^r\_x\; f\; =\; j^r\_x\; g\; \backslash Rightarrow\; F(f)|\_\; =\; F(g)|\_.$$Natural bundles of order r coincide with the associated fibre bundles to the r -th order frame bundles F ^{s(M)} .

A classical result by Epstein and Thurston shows that all natural bundles have finite order.^[1]

Examples

TM of a manifold M .

Other examples include the cotangent bundles, the bundles of metrics of signature

(r,s) and the bundle of linear connections.^[2]

Notes and References

1. Epstein . D. B. A. . David B. A. Epstein . Thurston . W. P. . William Thurston . 1979 . Transformation Groups and Natural Bundles . . en . s3-38 . 2 . 219–236 . 10.1112/plms/s3-38.2.219.
2. Book: Fatibene, Lorenzo . Natural and Gauge Natural Formalism for Classical Field Theorie . Francaviglia . Mauro . Mauro Francaviglia . 2003 . Springer . 978-1-4020-1703-2 . en . 10.1007/978-94-017-2384-8.