Natural bundle

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 ${\displaystyle p:F\to \mathrm {id} _{{\mathcal {M}}f_{n}}}$.

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]

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]

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]