synthetic differential geometry
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from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The Steenrod-approximation theorem states mild conditions under which an extension of a smooth function on a closed subset by a continuous function may itself be improved to an extension by a smooth function.
This is a smooth enhancement of the Tietze extension theorem.
Let be a finite dimensional connected smooth manifold with corners. Let be a locally trivial smooth bundle with a locally convex manifold as typical fiber and a continuous section.
If is a closed subset and is an open subset such that is smooth in a neighbourhood of , then:
for each open neighbourhood of in there exists a section
which is smooth in a neighbourhood of ;
and which equals on ;
there exists a homotopy between and such that
each is a section of ;
for we have .
See (Wockel)
Let be two smooth functions between smooth manifolds. Let be a continuous delayed homotopy between them, constant in a neighbourhood .
Then there exists also smooth homotopy between and which is itself continuously homotopic to .
To apply the generalized Steenrod theorem with the notation as stated there, make the following identifications
set ;
set ;
let be the trivial -bundle over
(so that sections of are equivalently functions )
let be the given continuous homotopy;
set ;
let .
Then because by assumption is a continuous delayed homotopy between smooth functions, it follows that is smooth in a neighbourhood of .
So the theorem applies and provides a smooth homotopy
which moroever is itself (continuously) homotopic to via some continuous .
Last revised on June 30, 2022 at 19:59:06. See the history of this page for a list of all contributions to it.