fibre bundle
Definition
Noun: 1. A topological structure: In mathematics, specifically in topology and differential geometry, a fibre bundle is a space that is locally a product space, but globally may have a different topological structure. It is defined by a continuous surjection from a total space to a base space, where the pre-image of each point in the base space (called the fibre) is homeomorphic to a fixed space. 2. A collection of filaments: In a biological or materials context, a fibre bundle refers to a group or bundle of fibres, such as nerve fibres (fascicles) or muscle fibres, bound together.
Usage Examples
- Mathematical Context:
- The tangent bundle of a manifold is a classic example of a fibre bundle.
- Understanding the concept of a fibre bundle is fundamental to modern theoretical physics.
- Biological/Anatomical Context:
- The optic nerve is composed of a fibre bundle carrying visual information to the brain.
- Under the microscope, you can see the muscle fibre bundle clearly.
Advanced Usage
- Principal Fibre Bundle: A fibre bundle where the fibre is a Lie group and the bundle has a continuous right action by that group which preserves the fibres.
- Vector Bundle: A specific and very important type of fibre bundle where each fibre is a vector space.
- Section of a Bundle: A continuous map from the base space back into the total space that selects one point from each fibre.
Variants and Related Words
- Fiber Bundle: The American English spelling.
- Bundle (noun): A collection of things or quantity of material tied or wrapped together. (This is the more general, non-technical term).
- Fibration (noun): A related topological concept, often used interchangeably with fibre bundle in a broader sense.
Synonyms
- Mathematical Context: (There are no perfect synonyms, but related concepts include) fibration, locally trivial bundle.
- Biological Context: fascicle, tract, bundle, strand.
Related Phrases
- Base Space: The underlying space over which the fibre bundle is defined.
- Total Space: The entire fibre bundle space.
- Fibre: The pre-image of a single point in the base space.
- Projection Map: The continuous surjection from the total space to the base space.
- Local Trivialization: The property that locally the bundle looks like a direct product of the base space and the fibre.
Noun
-
a bundle of fibers (especially nerve fibers)
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