nLab Whitney extension theorem

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A statement about sufficient data for extensions of a smooth function from a compact subset to an open neighbourhood.

extension theoremscontinuous functionssmooth functions
plain functionsTietze extension theoremWhitney extension theorem
equivariant functionsequivariant Tietze extension theorem

References

The original article is

  • Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, 1933 (pdf)

Textbook accounts include

  • Lars Hörmander, theore 2.3.6 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

Enhancement to a linear splitting of restriction maps on Fréchet spaces of sections with compact support of vector bundles:

This is then used to show the restriction map to (suitable) regular closed subsets is a submersion of mapping spaces (with maps valued in an arbitrary manifold).

See also

Last revised on June 30, 2022 at 19:55:45. See the history of this page for a list of all contributions to it.