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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 ...
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.
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 ...
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport.
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 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 ...