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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.
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 ...
Most known non-trivial computations of principal -connections are done with homogeneous spaces because of the triviality of the (co)tangent bundle. (For example, let /, be a principal -bundle over /) This means that 1-forms on the total space are canonically isomorphic to (,), where is the dual lie algebra, hence -connections are in bijection with (,).
A commonly cited example is the Dwork construction of the Picard–Fuchs equation.Let (,,) be the elliptic curve + + =.Here, is a free parameter describing the curve; it is an element of the complex projective line (the family of hypersurfaces in dimensions of degree n, defined analogously, has been intensively studied in recent years, in connection with the modularity theorem and its ...
If M is a surface in R 3, it is easy to see that M has a natural affine connection. From the linear connection point of view, the covariant derivative of a vector field is defined by differentiating the vector field, viewed as a map from M to R 3, and then projecting the result orthogonally back onto the tangent spaces of M. It is easy to see ...
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.