Support of a 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 support of $s$ is the closed set

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

where $0_x$ is the zero vector in $E_x$ . Equivalently, $x\notin\operatorname{supp}(s)$ exactly when $s$ vanishes identically on some neighborhood of $x$ . The section is compactly supported when $\operatorname{supp}(s)$ is compact; the space of such sections is denoted $\Gamma_c^\infty(M,E)$ .

Local description

In a local frame $e_1,\ldots,e_r$ , write $s=\sum_i s^i e_i$ . Then

\operatorname{supp}(s)\cap U = \bigcup_{i=1}^r\bigl(\operatorname{supp}(s^i)\cap U\bigr).

Hence support is independent of the chosen frame. The closure in the definition matters: the nonzero locus is open, but points where nonzero values accumulate belong to the support even when the section itself vanishes there.

Basic operations

For smooth sections $s,t$ and a smooth function $f$ ,

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

These inclusions can be strict because of cancellation or extra zeros. A vector bundle morphism over $M$ cannot enlarge support: $\operatorname{supp}(\Phi\circ s)\subseteq\operatorname{supp}(s)$ .

Compact support makes local constructions global. If a section is supported in a trivializing open set, it can be represented by compactly supported component functions there and extended by zero outside that open set, provided its support stays away from the boundary.

Conventions and scope

Some authors call $\{x:s(x)\neq0_x\}$ the support before taking its closure, but the closed-support convention used here is standard in analysis and differential geometry. The support is a subset of the base $M$ , not the image $s(M)\subseteq E$ . For sections of affine or general fiber bundles, “nonzero” requires a separately chosen reference section and is therefore not intrinsic.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 2, supports, bump functions, and partitions of unity.
L. W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: Chapter 13, compact supports and integration.