Bump function - Imported Knowlpedia Content

Definition

Let $M$ be a smooth manifold. A bump function is a smooth map $\varphi:M\to\mathbb R$ whose support

\operatorname{supp}\varphi=\overline{\{p\in M:\varphi(p)\ne0\}}

is a compact set. In localized constructions one usually requires more: for a closed set $A\subseteq M$ and an open set $U\supseteq A$ , a bump function for $A$ supported in $U$ satisfies $0\le\varphi\le1$ , equals $1$ on a neighborhood of $A$ , and has $\operatorname{supp}\varphi\subseteq U$ .

Existence on smooth manifolds

If $A\subseteq U\subseteq M$ , with $A$ closed and $U$ open, and $A$ is compact, then such a function exists. The proof first constructs Euclidean cutoff functions and then combines local constructions using a partition of unity. This localization theorem is developed in Lee, chapter on smooth manifolds and Nestruev, “Cutoff and Other Special Smooth Functions”.

Uses and constructions

Bump functions extend locally defined objects, splice constructions without changing them on a chosen region, and build partitions of unity. If $s$ is a local section over $U$ and $\varphi$ has support contained in $U$ , then $\varphi s$ , extended by zero, is a global smooth section.

Conventions and non-examples

Some authors use “cutoff function” for the more specific $0$ -to- $1$ function adapted to $A\subseteq U$ , and others do not require every cutoff to have compact support. The constant function $1$ is therefore a bump function precisely when $M$ is compact. A compactly supported continuous function with a corner is not a smooth bump function.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth functions, bump functions, and partitions of unity.
Jet Nestruev, Smooth Manifolds and Observables, Springer, 2003. DOI record. Relevant: “Cutoff and Other Special Smooth Functions.”