Natural bundle

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 $s\ge 1$. 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 $M{f}_n$ 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. $B(M)$ is a fibred manifold over $M$, with projection denoted by $p_M:B(M)\to M$;
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 $p:F\to B$.

Finite order natural bundles

A natural bundle $F:M{f}_n\to Mf$ 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 $j_x^{r}f$. 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 $F^s(M)$.

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

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.

References

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