nLab natural bundle





Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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 TXT X of a manifold XX. For f:XYf \colon X \to Y any smooth function then push-forward of vector fields yields a fiberwise linear morphism df:TXTYd f \colon T X \to T Y of tangent bundles which covers ff, in that the diagram

TX df TY X f Y \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.



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:

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]

Last revised on August 18, 2023 at 14:03:21. See the history of this page for a list of all contributions to it.