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An affine frame for A consists of a point p ∈ A and a basis (e 1,… e n) of the vector space T p A = R n. The general linear group GL(n) acts freely on the set FA of all affine frames by fixing p and transforming the basis (e 1,… e n) in the usual way, and the map π sending an affine frame (p; e 1,… e n) to p is the quotient map.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
If the section s is replaced by a new section sg, defined by (sg)(x) = s(x)g(x), where g:M→G is a smooth map, then () = +. The principal connection is uniquely determined by this family of g {\displaystyle {\mathfrak {g}}} -valued 1-forms, and these 1-forms are also called connection forms or connection 1-forms , particularly in older or ...
A manifold having a distinguished affine structure is called an affine manifold and the charts which are affinely related to those of the affine structure are called affine charts. In each affine coordinate domain the coordinate vector fields form a parallelisation of that domain, so there is an associated connection on each domain.
Strong coupling interactions are most well known to model synchronization effects of dynamic spatial systems such as the Kuramoto model. These classifications do not reflect the local or global (GMLs [11]) coupling nature of the interaction. Nor do they consider the frequency of the coupling which can exist as a degree of freedom in the system ...
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 .
Near to any point, S can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. The affine subspaces are model surfaces—they are the simplest surfaces in R 3, and are homogeneous under the Euclidean group of the plane, hence they are Klein geometries in the sense of Felix Klein's Erlangen programme.
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.