Zero section - Knowlpedia

Let $\pi:E\to M$ be a smooth vector bundle. Its zero section is the map

0_E:M\longrightarrow E, \qquad 0_E(x)=0_x\in E_x.

It is a smooth section because every vector-bundle trivialization identifies it with $x\mapsto(x,0)$ . The identity $\pi\circ0_E=\operatorname{id}_M$ makes $0_E$ injective, and it is a smooth embedding. Its image, also called the zero section, is the embedded copy $\{0_x:x\in M\}\subseteq E$ . No trivialization, connection, or metric is required for this construction.

Canonical properties

Every vector bundle morphism $\Phi:E\to F$ covering $f:M\to N$ preserves zero vectors:

\Phi\circ0_E=0_F\circ f.

Thus the zero section is natural with respect to vector bundle morphisms. It is also the additive identity in the module of smooth sections.

Tangent and normal geometry

Along $0_E(M)$ , the differential of the projection splits the tangent bundle:

TE|_{0_E(M)} \cong T M\oplus E.

The first summand is tangent to the embedded zero section, and the second consists of vertical tangent vectors in the fibers. Consequently the normal bundle of $0_E(M)\subseteq E$ is canonically isomorphic to $E$ . This canonical identification is frequently used in tubular-neighborhood constructions and in definitions of the Thom class.

Examples and warning

For the tangent bundle $TM\to M$ , the zero section sends $x$ to the zero tangent vector $0\in T_xM$ . For a trivial bundle $M\times\mathbb F^r$ , it is $x\mapsto(x,0)$ .

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 10, vector bundles, sections, and embedded zero sections.
D. Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: Chapter 3, vector-bundle constructions.