Search results
Results from the WOW.Com Content Network
In other words, is fiber-preserving, and f is the induced map on the space of fibers of E: since π E is surjective, f is uniquely determined by . For a given f , such a bundle map φ {\displaystyle \varphi } is said to be a bundle map covering f .
Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney. [11] Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, [12] that is a fiber bundle whose fiber is a sphere of arbitrary dimension. [13]
FICON (Fibre Connection) is the IBM proprietary name for the ANSI FC-SB-3 Single-Byte Command Code Sets-3 Mapping Protocol for Fibre Channel (FC) protocol.It is a FC layer 4 protocol used to map both IBM's antecedent (either ESCON or parallel Bus and Tag) channel-to-control-unit cabling infrastructure and protocol onto standard FC services and infrastructure.
In this way, the connection form can be used to define the horizontal bundle: The horizontal bundle is the kernel of the connection form. The solder form or tautological one-form vanishes on the vertical bundle and is non-zero only on the horizontal bundle. By definition, the solder form takes its values entirely in the horizontal bundle.
Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X × B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
In summary, there is a one-to-one correspondence (up to equivalence) between the descents of principal connections to associated fiber bundles, and G-connections on associated fiber bundles. For this reason, in the category of fiber bundles with a structure group G , the principal connection contains all relevant information for G -connections ...
where R E is the scalar curvature on the bundle as a whole (obtained from the metric π * g+kω above), and R M (g) is the scalar curvature on the base manifold M (the Lagrangian density of the Einstein–Hilbert action), and L(g, ω) is the Lagrangian density for the Yang–Mills action, and R G (k) is the scalar curvature on each fibre ...