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
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For a smooth function between smooth manifold, and for a differential n-form, there is the pullback -form .
If differential forms are defined as linear duals to vectors then pullback is the dual operation to pushforward of a vector field
Differential forms may be defined by Yoneda extension from differential forms on Cartesian spaces (see at geometry of physics – differential forms).
For and Cartesian spaces and a smooth function between them, and on differential 1-forms
the pullback operation is given by
and hence
where
is the pullback of functions defined by
the function
is the partial derivative of the -th coordinate component of along the the coordinate.
(compatiblity with the de Rham differential)
Pullback of differential forms commutes with the de Rham differential:
Hence it constitutes a chain map between the de Rham complexes
Under pullback differential forms form a presheaf on the catories CartSp and SmthMfd, in fact a sheaf with respect to the standard open cover-coverage.
Notions of pullback:
pullback, fiber product (limit over a cospan)
lax pullback, comma object (lax limit over a cospan)
(∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)
A standard reference is
See also for instance
Last revised on June 21, 2024 at 09:35:33. See the history of this page for a list of all contributions to it.