# nLab natural bundle

Contents

bundles

## Constructions

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

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]