nLab
bump function

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Higher geometry

Contents

Idea

A bump function is a smooth function with compact support, especially one that is not zero on a space that it not compact.

One reason the category SmthMfd of smooth manifolds and smooth functions is important is that a good supply of bump functions exists; this fact accounts for the great flexibility of smooth objects, in stark contrast to analytic geometry or algebraic varieties (since non-zero analytic or algebraic functions with compact support exist only on compact spaces). For instance, bump functions can be used to construct partitions of unity, which can in turn be used to smoothly patch together local structures into a global structure without obstruction, as for example Riemannian metrics.

Constructions

Define a function ϕ(x)\phi(x) on the standard open unit ball of the cartesian space n\mathbb{R}^n by

ϕ(x)=exp(1x 21) \phi(x) = \exp\left( \frac1{{\|x\|^2} - 1} \right)

so that ϕ(x)\phi(x) and all of its higher derivatives vanish rapidly as xx approaches the boundary. This gives a smooth function compactly supported on the unit ball? centered at the origin.

For ε>0\varepsilon \gt 0, define ϕ ε(x)ϕ(x/ε)\phi_\varepsilon(x) \coloneqq \phi(x/\varepsilon), and define

ψ ε=ϕ εϕ ε 1\psi_\varepsilon = \frac{\phi_\varepsilon}{{\|\phi_\varepsilon\|}_1}

so that the family (ψ ε) ε(\psi_\varepsilon)_\varepsilon is a smooth approximation to the identity? (of convolution, the Dirac functional), compactly supported on the closed ball of radius ε\varepsilon and having an L 1L^1-norm equal to 11. Then, it is standard that for any pair KVK \subset V with KK compact and VV open in a cartesian space n\mathbb{R}^n, we can choose an open UU containing KK and with compact closure contained in VV, and then taking the convolution product

ψ ε*χ U\psi_\varepsilon \ast \chi_U

of ψ ε\psi_\varepsilon with the characteristic function χ U\chi_U, for ε\varepsilon sufficiently small, we obtain a smooth function equal to 11 on KK and equal to 00 outside VV.

By performing the above construction in charts, we obtain, in any smooth manifold, a smooth function equal to 11 on any compact subspace KK and equal to 00 outside any neighbourhood VV of KK. (This is a smooth regularity property.)

References

Revised on December 6, 2011 05:22:44 by Toby Bartels (64.89.53.227)