Vector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space (for example could be a topological space, a manifold, or an algebraic variety): to every point of the space we associate (or "attach") a vector space in such a way that these vector spaces fit together to form another space of the same kind as (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over .
Fiber bundleIn mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle.
Principal bundleIn mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with An action of on , analogous to for a product space. A projection onto . For a product space, this is just the projection onto the first factor, . Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of .
Bundle mapIn mathematics, a bundle map (or bundle morphism) is a morphism in the of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the . Then in the fourth section, some other examples will be given.
Associated bundleIn mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . For a fiber bundle F with structure group G, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ.
Frame bundleIn mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal GL(k, R)-bundle (where k is the rank of E). The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle.
Line bundleIn mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner.
Section (fiber bundle)In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, : then a section of that fiber bundle is a continuous map, such that for all . A section is an abstract characterization of what it means to be a graph. The graph of a function can be identified with a function taking its values in the Cartesian product , of and : Let be the projection onto the first factor: .
Fibre-reinforced plasticFibre-reinforced plastic (FRP; also called fibre-reinforced polymer, or in American English fiber) is a composite material made of a polymer matrix reinforced with fibres. The fibres are usually glass (in fibreglass), carbon (in carbon-fibre-reinforced polymer), aramid, or basalt. Rarely, other fibres such as paper, wood, boron, or asbestos have been used. The polymer is usually an epoxy, vinyl ester, or polyester thermosetting plastic, though phenol formaldehyde resins are still in use.
Tautological bundleIn mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles.
