vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
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 $T X$ 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
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:
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:
Last revised on August 18, 2023 at 14:03:21. See the history of this page for a list of all contributions to it.