nLab natural bundle

Idea

A type of fiber bundle (usually: vector bundle) over (smooth) manifolds is called natural if suitable homomorphisms of manifolds naturally lift to morphisms of the corresponding bundles covering them.

The archetypical example is the tangent bundle $T X$ of a manifold $X$. For $f \colon X \to Y$ any smooth function then push-forward of vector fields yields a fiberwise linear morphism $d f \colon T X \to T Y$ of tangent bundles which covers $f$, in that the diagram

$$\array{ T X &\stackrel{d f}{\longrightarrow}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\longrightarrow}& Y }$$

commutes. (This functoriality of the tangent bundle construction is incidentally also the incarnation of the chain rule (see there) of differentiation).

Formally, there is a category of fiber bundles over manifolds with morphisms the morphisms of bundles covering morphisms of the base spaces, and there is a projection functor from fiber bundles to their base manifolds. A natural bundle is a section of this projection functor.

Also “contravariantly natural” bundles such as bundle of covectors are sometimes referred to as natural bundles (e.g. Kolář-Michor-Slovak). But these are covariantly natural not with respect to all smooth functions between manifolds, but only for local diffeomorphisms.

Examples

• tractor bundle

• Schouten-Nijenhuis bracket

References

Original discussion:

• Albert Nijenhuis, Geometric aspects of formal differential operations on tensor fields, Proceedings of the International Congress of Mathematicians 1958, 463–469. PDF.

• Albert Nijenhuis, Natural bundles and their general properties, in: Differential Geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 317–334. PDF.

• Chuu-Lian Terng, Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1978) 775-828 [doi:10.2307/2373910, jstor:2373910]

Lecture notes and textbook accounts:

• Demeter Krupka, Josef Janyška, Part 2 of: Lectures on differential invariants, Univerzita J. E. Purkyně, Brno (1990) [ISBN:80-210-165-8, researchgate]

• Ivan Kolář, Peter Michor, Jan Slovák, section 14 of: Natural operations in differential geometry, Springer (1993) [webpage, doi:10.1007/978-3-662-02950-3, pdf]

The example of the Schouten-Nijenhuis bracket:

• Peter W. Michor, Remarks on the Schouten-Nijenhuis bracket, In: Bureš, J. and Souček, V. (eds.): Proceedings of the Winter School “Geometry and Physics” Circolo Matematico di Palermo, Palermo (1987) 207-215 [dml:701423, pdf, pdf]