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It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations . In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations .
The pair (P, η) defines the structure of an affine geometry on M, making it into an affine manifold. The affine Lie algebra aff(n) splits as a semidirect product of R n and gl(n) and so η may be written as a pair (θ, ω) where θ takes values in R n and ω takes values in gl(n).
If the principal bundle P is the frame bundle, or (more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form θ, which is an equivariant R n-valued 1-form on P, should be taken into account.
Malvin Ruderman and Charles Kittel of the University of California, Berkeley first proposed the model to explain unusually broad nuclear spin resonance lines in natural metallic silver. The theory is an indirect exchange coupling : the hyperfine interaction couples the nuclear spin of one atom to a conduction electron also coupled to the spin ...
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 ...
An affine bundle is a fiber bundle with a general affine structure group (,) of affine transformations of its typical fiber of dimension . This structure group always is reducible to a general linear group G L ( m , R ) {\displaystyle GL(m,\mathbb {R} )} , i.e., an affine bundle admits an atlas with linear transition functions.
In the term mode coupling, as used in physics and electrical engineering, the word "mode" refers to eigenmodes of an idealized, "unperturbed", linear system. The superposition principle says that eigenmodes of linear systems are independent of each other: it is possible to excite or to annihilate a specific mode without influencing any other ...
Shape-memory coupling is a system for connecting pipes using shape-memory alloys. In its typical form the technique uses an internally ribbed sleeve of alloy such as Tinel (see Nitinol ) that is slightly smaller in diameter than the pipes it is to connect.