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Given a vector bundle of rank , and any representation : (,) into a linear group (), there is an induced connection on the associated vector bundle =. This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of E {\displaystyle E} and using the theory of principal bundles.
The definition may be phrased for a connection on a vector bundle or principal bundle, with the two perspectives being essentially interchangeable. Here the definition of principal bundles is presented, which is the form that appears in Hitchin's work. [1] [5] [6]
The following is a list of Mac software – notable computer applications for current macOS operating systems. For software designed for the Classic Mac OS , see List of old Macintosh software . Audio software
If is a vector bundle, there is one-to-one correspondence between linear connections on and the connections on the ()-module of sections of . Strictly speaking, ∇ {\displaystyle \nabla } corresponds to the covariant differential of a connection on E → X {\displaystyle E\to X} .
System Information, formerly System Profiler, is a software utility derived from field service diagnostics produced by Apple's Service Diagnostic Engineering team, at that time located in Apple satellite buildings in Campbell, California, that was bundled with the classic Mac OS since Mac OS 7.6 under the name Apple System Profiler.
Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J 1 Y of the jet bundle J 1 Y → Y of Y is an affine bundle morphism over X. In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X. (That is, the ...
If one has a vector bundle E over M, then the metric can be extended to the entire vector bundle, as the bundle metric. One may then define a connection that is compatible with this bundle metric, this is the metric connection. For the special case of E being the tangent bundle TM, the metric connection is called the Riemannian connection.
Grand Central Dispatch (GCD or libdispatch) is a technology developed by Apple Inc. to optimize application support for systems with multi-core processors and other symmetric multiprocessing systems. [2] It is an implementation of task parallelism based on the thread pool pattern. The fundamental idea is to move the management of the thread ...