Natural bundle

 ( Differential Geometry ) 

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 $s\; \backslash geq\; 1$. 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.

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Definition Let $Mf$ denote the category of smooth manifolds and Smooth map and $Mf\_n$ the category of smooth $n$-dimensional manifolds and local diffeomorphisms. Consider also the category $\backslash mathcal\{FM\}$ of Fibered manifold and bundle morphisms, and the functor $B:\; \backslash mathcal\{FM\}\; \backslash to\; \backslash mathcal\{M\}f$ associating to any fibred manifold its base manifold.

A natural bundle (or bundle functor) is a functor $F:\; \backslash mathcal\{M\}f\_n\; \backslash to\; \backslash mathcal\{FM\}$ satisfying the following three properties:

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

As a consequence of the first condition, one has a natural transformation $p:\; F\; \backslash to\; B$.

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Finite order natural bundles A natural bundle $F:\; Mf\_n\; \backslash to\; Mf$ is called of finite order $r$ if, for every local diffeomorphism $f:\; M\; \backslash to\; N$ and every point $x\; \backslash in\; M$, the map $F(f)\_x:\; F(M)\_\{x\}\; \backslash to\; F(N)\_\{f(x)\}$ depends only on the jet $j^r\_x\; f$. Equivalently, for every local diffeomorphisms $f,g:\; M\; \backslash to\; N$ and every point $x\; \backslash in\; M$, one has$$j^r\_x\; f\; =\; j^r\_x\; g\; \backslash Rightarrow\; F(f)|\_\{F(M)\_x\}\; =\; F(g)|\_\{F(M)\_x\}.$$Natural bundles of order $r$ coincide with the associated fibre bundles to the $r$-th order Frame bundle $F^s(M)$.

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

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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.

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Notes

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