Compactly supported section - Imported Knowlpedia Content

Definition

Let $E\to M$ be a smooth vector bundle and $s\in\Gamma^\infty(M,E)$ a smooth section. The section $s$ is compactly supported if its support

\operatorname{supp}(s)=\overline{\{x\in M\mid s(x)\neq 0_x\}}

is a compact subset of $M$ . The vector space of compactly supported smooth sections is denoted

\Gamma_c^\infty(M,E) =\{s\in\Gamma^\infty(M,E)\mid \operatorname{supp}(s)\text{ is compact}\}.

Compactness is taken in the topology of the base, so the definition is intrinsic and does not depend on a bundle trivialization, metric, or connection.

Basic properties

If $s,t\in\Gamma_c^\infty(M,E)$ and $f\in C^\infty(M)$ , then $s+t$ and $fs$ are compactly supported because

\operatorname{supp}(s+t)\subseteq\operatorname{supp}(s)\cup\operatorname{supp}(t), \qquad \operatorname{supp}(fs)\subseteq\operatorname{supp}(s).

Thus $\Gamma_c^\infty(M,E)$ is a module over $C^\infty(M)$ . It is also preserved by vector-bundle morphisms over $M$ .

When $M$ is compact, every smooth section is compactly supported. On a noncompact base, compact support is a genuine restriction and is the natural condition for integration by parts without boundary terms at infinity.

Local construction

Let $U\subseteq M$ be a trivializing open set and choose a smooth function $\chi$ with compact support contained in $U$ . Multiplying a section on $U$ by $\chi$ and extending it by zero produces a member of $\Gamma_c^\infty(M,E)$ . Partitions of unity therefore reduce many global constructions involving compactly supported sections to finitely many local ones near their common compact support.

Examples and non-examples

For the trivial line bundle over $\mathbb R$ , a bump function is a compactly supported section. The function $x\mapsto e^{-x^2}$ is not compactly supported: rapid decay does not replace the requirement that the section vanish outside a compact set.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 2, supports, bump functions, and partitions of unity.
Lars Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd ed., Springer, 1990. DOI record. Relevant: Chapter 1, test functions and compact support.