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.

Noun

1. a bundle of fibers (especially nerve fibers)