Natural bundle

In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the higher order frame bundle $F^r(M)$, for some $r\ge 1$. In other words, 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 $r$.

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.

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 $ℱℳ$ of fibred manifolds and bundle morphisms, and the functor $B:ℱℳ\to ℳf$ associating to any fibred manifold its base manifold.

A natural bundle (or bundle functor) is a functor $F:ℳf_n\to ℱℳ$ satisfying the following three properties:

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

As a consequence of the first condition, one has a natural transformation $p:F\to {\mathrm{i}\mathrm{d}}_{ℳf_n}$.

Finite order natural bundles

A natural bundle $F:ℳf_n\to ℱℳ$ 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 $j_x^{r}f$. Equivalently, for every local diffeomorphisms $f,g:M\to N$ and every point $x\in M$, one has$$j_x^{r}f=j_x^{r}g\Rightarrow F(f){|}_{F(M{)}_x}=F(g){|}_{F(M{)}_x}.$$Natural bundles of order $r$ coincide with the associated fibre bundles to the $r$-th order frame bundles $F^r(M)$.

After various intermediate cases, it was proved by Epstein and Thurston that all natural bundles have finite order.

Natural $\Gamma$-bundles

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

Under suitable assumptions, natural $\Gamma$-bundles have finite order as well.

Examples

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

Other examples include the cotangent bundles, the bundles of metrics of signature $(r,s)$ and the bundle of linear connections.