fibre bundle

Thân thiện

A scientist carefully examines a fibre bundle under a microscope.

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.

Học thuật

Noun

1. a bundle of fibers (especially nerve fibers)

Từ đồng nghĩa

• fiber bundle
• fascicle
• fasciculus