fiber bundle

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fiber bundle

A scientist examines a fiber bundle under a bright light.

Definition
  1. Noun:
    • A topological structure: In mathematics, specifically in topology and geometry, a 'fiber bundle' is a space that is locally a product space but may have a different global topological structure. It consists of a base space, a total space, and a projection map.
    • A bundle of fibers: In a more general or anatomical sense, a 'fiber bundle' can refer to a collection of fibers, such as nerve fibers or muscle fibers, bound together.
Usage Examples
  • Noun (Mathematical Context):
    • The tangent bundle of a manifold is a classic example of a fiber bundle.
    • Understanding the concept of a fiber bundle is fundamental in differential geometry.
  • Noun (Anatomical/Biological Context):
    • The optic nerve is a fiber bundle that transmits visual information to the brain.
    • Damage to the nerve fiber bundle can result in loss of motor function.
Advanced Usage
  • "Principal fiber bundle": A fiber bundle where the fiber is a Lie group acting on the total space.
    • Gauge theories in physics are often formulated using principal fiber bundles.
  • "Vector bundle": A fiber bundle where each fiber is a vector space.
    • A line bundle is a simple type of vector bundle with one-dimensional fibers.
Variants and Related Words
  • Fibre bundle: (noun) The British English spelling of 'fiber bundle'.
  • Bundle: (noun) A collection of things or quantity of material tied or wrapped together.
  • Fibration: (noun) A continuous surjection which, intuitively, is a generalization of a fiber bundle.
Synonyms
  • Fibre bundle (UK spelling).
  • In anatomy/biology: nerve tract, fascicle, fasciculus.
Related Phrases
  • Fiber bundle projection: The continuous surjective map from the total space to the base space in a fiber bundle.
    • The fiber bundle projection must satisfy the local triviality condition.
  • Section of a fiber bundle: A continuous right inverse of the projection map.
    • A nowhere-zero section of a vector bundle implies the bundle is trivial.
Related Concepts
  • Local triviality: The property that defines a fiber bundle; locally, it looks like a direct product of the base and a fiber.
    • The local triviality of the fiber bundle allows for coordinate representations.
  • Transition functions: Functions that describe how the local product representations of a fiber bundle are glued together.
    • The topology of the fiber bundle is encoded in its transition functions.
fiber bundle

A scientist examines a fiber bundle under a bright light.

Noun
  1. a bundle of fibers (especially nerve fibers)

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