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section of a fiber bundle
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(Definition)
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Let $\funcdef{p}{E}{B}$ be a fiber bundle, denoted by $\xi.$
A section of $\xi$ is a continuous map $\funcdef{s}{B}{E}$ such that the composition $p\comp s$ equals the identity. That is, for every $b\in B,$ $s(b)$ is an element of the fiber over $b.$ More generally, given a topological subspace $A$ of $B,$ a section of $\xi$
over $A$ is a section of the restricted bundle $\funcdef{\restr{p}{A}}{\inv{p}(A)}{A}.$
The set of sections of $\xi$ over $A$ is often denoted by $\Gamma(A;\xi),$ or by $\Gamma(\xi)$ for sections defined on all of $B.$ Elements of $\Gamma(\xi)$ are sometimes called global sections, in contrast with the local sections $\Gamma(U;\xi)$ defined on an open set $U.$
Remark 1 If $E$ and $B$ have, for example, smooth structures, one can talk about smooth sections of the bundle. According to the context, the notation $\Gamma(\xi)$ often denotes smooth sections, or some other set of suitably restricted sections.
Example 1 If $\xi$ is a trivial fiber bundle with fiber $F,$ so that $E=F\cross B$ and $p$ is projection to $B,$ then sections of $\xi$ are in a natural bijective correspondence with continuous functions $\funcsig{B}{F}.$
Example 2 If $B$ is a smooth manifold and $E=TB$ its tangent bundle, a ( smooth) section of this bundle is precisely a (smooth) tangent vector field.
In fact, any tensor field on a smooth manifold $M$ is a section of an appropriate vector bundle. For instance, a contravariant $k$ -tensor field is a section of the bundle $TM^{\otimes k}$ obtained by repeated tensor product from the tangent bundle, and similarly for covariant and mixed tensor fields.
Example 3 If $B$ is a smooth manifold which is smoothly embedded in a Riemannian manifold $M,$ we can let the fiber over $b\in B$ be the orthogonal complement in $T_b M$ of the tangent space $T_b B$ of $B$ at $b$ . These choices of fiber turn out to make up a vector bundle $\nu(B)$ over $B,$ called the normal bundle of $B$ . A section of $\nu(B)$ is a normal vector field on $B.$
Example 4 If $\xi$ is a vector bundle, the zero section is defined simply by $s(b)=0,$ the zero vector on the fiber.
It is interesting to ask if a vector bundle admits a section which is nowhere zero. The answer is yes, for example, in the case of a trivial vector bundle, but in general it depends on the topology of the spaces involved. A well-known case of this question is the hairy ball theorem, which says that there are no nonvanishing tangent vector fields on the sphere.
Example 5 If $\xi$ is a principal $G$ - bundle, the existence of any section is equivalent to the bundle being trivial.
Remark 2 The correspondence taking an open set $U$ in $B$ to $\Gamma(U;\xi)$ is an example of a sheaf on $B.$
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"section of a fiber bundle" is owned by antonio.
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See Also: fiber bundle
| Other names: |
section, cross section, cross-section |
| Also defines: |
smooth section, global section, local section, zero section |
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Cross-references: sheaf, sphere, hairy ball theorem, topology, trivial vector bundle, zero vector, normal vector, orthogonal complement, Riemannian manifold, tensor product, vector bundle, tensor, field, tangent vector, smooth, tangent bundle, smooth manifold, bijective, smooth structures, open set, topological subspace, fiber, identity, composition, continuous map, fiber bundle
There are 32 references to this entry.
This is version 7 of section of a fiber bundle, born on 2003-02-10, modified 2004-06-19.
Object id is 4008, canonical name is SectionOfAFiberBundle.
Accessed 14730 times total.
Classification:
| AMS MSC: | 55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles) |
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Pending Errata and Addenda
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