Nowhere-vanishing section - Imported Knowlpedia Content

Definition

Let $\pi:E\to M$ be a real or complex vector bundle and let $s:M\to E$ be a section. The section $s$ is nowhere vanishing if

s(x)\neq 0_x

for every $x\in M$ , where $0_x$ is the value of the zero section in $E_x$ . Equivalently, the image $s(M)$ is disjoint from the image of the zero section. In the smooth category, $s$ is required to be smooth; in the topological category, it is required to be continuous. No choice of norm is needed because nonzeroness is intrinsic to each vector space $E_x$ .

Trivial line summand

A nowhere-vanishing section spans a rank-one subbundle

L_s=\{(x,\lambda s(x))\mid x\in M,\ \lambda\in\mathbb F\}\subseteq E,

canonically trivialized by $s$ . After choosing a bundle metric, the orthogonal complement gives a splitting

E\cong \underline{\mathbb F}\oplus L_s^\perp.

Conversely, a trivial line subbundle of $E$ supplies a nowhere-vanishing section. Thus existence of such a section is equivalent to splitting off a trivial line summand, although the complementary summand is not canonical.

Obstructions

For an oriented real rank- $n$ vector bundle, the Euler class is the primary obstruction to a nowhere-vanishing section. A nowhere-vanishing section forces the Euler class to vanish. The converse requires additional hypotheses and is not true as an unrestricted statement: higher obstruction classes can remain when the base has dimension greater than the rank Milnor–Stasheff, §12.

Examples and non-examples

Every trivial vector bundle of positive rank has a constant nowhere-vanishing section. The tangent bundle of $S^1$ has one, while the tangent bundle of $S^2$ has none by the hairy-ball theorem. A section that vanishes at even one point is a decisive non-example, regardless of whether its zero is isolated.

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: Chapter 3, sections and trivial summands of vector bundles.
John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. Publisher record. Relevant: §12, the Euler class as an obstruction to a nonzero section.