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 $s \geq 1$ . 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 $s$ .{{citation|title=Natural bundles have finite order|last=Palais|first=Richard|author-link=Richard Palais |last2=Terng |first2=Chuu-Lian |author-link2= Chuu-Lian Terng|journal= Topology|volume=16|pages=271–277|year=1977|doi=10.1016/0040-9383(77)90008-8|hdl=10338.dmlcz/102222|hdl-access=free}}

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.{{citation|title=Natural bundles and their general properties|author=A. Nijenhuis |publisher=Diff. Geom. in Honour of K. Yano|year=1972|pages=317–334|location= Tokyo|author-link=Albert Nijenhuis}}

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 $\mathcal{FM}$ of fibred manifolds and bundle morphisms, and the functor $B: \mathcal{FM} \to \mathcal{M}f$ associating to any fibred manifold its base manifold.

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

$B \circ F = \mathrm{id}$ , i.e. $B(M)$ is a fibred manifold over $M$ , with projection denoted by $p_M: B(M) \to M$ ;
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 $p^{-1}(U) \hookrightarrow F(M)$ ;
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 $F: Mf_n \to Mf$ 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 $j^r_x f$ . Equivalently, for every local diffeomorphisms $f,g: M \to N$ and every point $x \in M$ , one has $j^r_x f = j^r_x g \Rightarrow F(f)|_{F(M)_x} = F(g)|_{F(M)_x}.$ 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.{{Cite journal |last=Epstein |first=D. B. A. |author-link=David B. A. Epstein |last2=Thurston |first2=W. P. |author-link2=William Thurston |date=1979 |title=Transformation Groups and Natural Bundles |url=http://doi.wiley.com/10.1112/plms/s3-38.2.219 |journal=Proceedings of the London Mathematical Society |language=en |volume=s3-38 |issue=2 |pages=219–236 |doi=10.1112/plms/s3-38.2.219}}

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.{{Cite book |last=Fatibene |first=Lorenzo |url=https://link.springer.com/book/10.1007/978-94-017-2384-8 |title=Natural and Gauge Natural Formalism for Classical Field Theorie |last2=Francaviglia |first2=Mauro |author-link2=Mauro Francaviglia |date=2003 |publisher=Springer |isbn=978-1-4020-1703-2 |language=en |doi=10.1007/978-94-017-2384-8}}

Notes

References

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Category:Differential geometry

Category:Manifolds

Category:Fiber bundles