Natural bundle

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

A natural bundle ${\displaystyle F:Mf_{n}\to Mf}$  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 ${\displaystyle F^{s}(M)}$ .

A classical result by Epstein and Thurston shows that all natural bundles have finite order.^[3]

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.^[4]

• 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