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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 ...
The fact that the Ehresmann connection is linear implies that in addition it verifies for every function on the Leibniz rule, i.e. () = + (), and therefore is a covariant derivative of s. Conversely a covariant derivative ∇ on a vector bundle defines a linear Ehresmann connection by defining H e , for e ∈ E with x = π ( e ), to be the ...
For any linear representation W of G there is an associated vector bundle over M, and a principal connection induces a covariant derivative on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of P × G W {\displaystyle P\times ^{G}W} over M is isomorphic to the space of G -equivariant W ...
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 ...
A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.
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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} .