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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).
Affine diagrams are denoted as (), (), or (), where X is the letter of the corresponding finite diagram, and the exponent depends on which series of affine diagrams they are in. The first of these, X l ( 1 ) , {\displaystyle X_{l}^{(1)},} are most common, and are called extended Dynkin diagrams and denoted with a tilde , and also sometimes ...
In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z, (x + y + z)/3, ix + (1 − i)y, etc.
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 ...
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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 .
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 ...
The limiting tangent line to the locus of midpoints as we approach p is exactly the affine normal line, i.e. the line containing the affine normal vector to γ(I) at γ(t 0). Notice that this is an affine invariant construction since parallelism and midpoints are invariant under affine transformations.