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In older literature, the term linear connection is occasionally used for an Ehresmann connection or Cartan connection on an arbitrary fiber bundle, [1] to emphasise that these connections are "linear in the horizontal direction" (i.e., the horizontal bundle is a vector subbundle of the tangent bundle of the fiber bundle), even if they are not ...
The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative , an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections ...
A G-connection on E is an Ehresmann connection such that the parallel transport map τ : F x → F x′ is given by a G-transformation of the fibers (over sufficiently nearby points x and x′ in M joined by a curve). [5] Given a principal connection on P, one obtains a G-connection on the associated fiber bundle E = P × G F via pullback.
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. Viewed infinitesimally, a connection is a way of differentiating geometric data in such a manner that the derivative is also geometrically meaningful.
Therefore, an affine connection is associated to a principal connection. It always exists. For any affine connection Γ : Y → J 1 Y, the corresponding linear derivative Γ : Y → J 1 Y of an affine morphism Γ defines a unique linear connection on a vector bundle Y → X. With respect to linear bundle coordinates (x λ, y i) on Y, this ...
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his ...
A linear connection ∇ on TM arising in this way is said to be compatible with Q. Connections compatible with Q are also called adapted connections. Concretely speaking, adapted connections can be understood in terms of a moving frame. [7] Suppose that V i is a basis of local sections of TM (i.e., a frame on M) which defines a section of Q ...
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties. The fibers of the vector bundle are the de Rham cohomology groups H D R k ( V s ) {\displaystyle H_{DR}^{k}(V_{s})} of the fibers V s {\displaystyle V_{s}} of the family.