Natural bundle

In differential geometry, a field in mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle ${\displaystyle F^{s}(M)}$ for some ${\displaystyle s\geq 1}$. It turns out that 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 ${\displaystyle s}$.^[1]

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.^[2]

Definition

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 ${\displaystyle p:F\to B}$.

Finite order natural bundles

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]

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.^[4]