Natural bundle

In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the higher order frame bundle ${\displaystyle F^{r}(M)}$, for some ${\displaystyle r\geq 1}$. In other words, its transition functions depend functionally on local changes of coordinates in the base manifold ${\displaystyle M}$ together with their partial derivatives up to order at most ${\displaystyle r}$.^[1][2]

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.^[3]

Definition

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

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

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

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

Finite order natural bundles

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

After various intermediate cases,^[1][4] it was proved by Epstein and Thurston that all natural bundles have finite order.^[2]

Natural ${\displaystyle \Gamma }$-bundles

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

Under suitable assumptions, natural ${\displaystyle \Gamma }$-bundles have finite order as well.^[5][6][7]

Examples

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

Other examples include the cotangent bundles, the bundles of metrics of signature ${\displaystyle (r,s)}$ and the bundle of linear connections.^[8]

Notes

1. Palais, Richard S.; Terng, Chuu-Lian (1977-01-01). "Natural bundles have finite order". Topology. 16 (3): 271–277. doi:10.1016/0040-9383(77)90008-8. ISSN 0040-9383.
2. 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.
3. Albert, Nijenhuis (1972). "Natural bundles and their general properties" (PDF). Differential Geometry (in honor of Kentaro Yano). Tokyo: Kinokuniya: 317–334.
4. Terng, Chuu Lian (1978). "Natural Vector Bundles and Natural Differential Operators". American Journal of Mathematics. 100 (4): 775–828. doi:10.2307/2373910. ISSN 0002-9327.
5. Slovák, Jan (1991). "Bundle functors on fibred manifolds". Annals of Global Analysis and Geometry. 9 (2): 129–143. doi:10.1007/BF00776852. ISSN 0232-704X.
6. Kolář, Ivan; Slovák, Jan; Michor, Peter W. (1993). Natural Operations in Differential Geometry. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-02950-3. ISBN 978-3-642-08149-1.
7. Benalili, Mohamed (1994-09-01). "Fibrés naturels sur la catégorie des Γ-variétés" [Natural bundles on the category of Γ-manifolds]. Rendiconti del Circolo Matematico di Palermo Series 2 (in French). 43 (3): 309–328. doi:10.1007/BF02844245. ISSN 1973-4409.
8. 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

• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag, archived from the original (PDF) on 2017-03-30, retrieved 2017-08-15
• Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
• Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7