Search results
Results From The WOW.Com Content Network
Given a vector bundle of rank , and any representation : (,) into a linear group (), there is an induced connection on the associated vector bundle =. This theory is most succinctly captured by passing to the principal bundle connection on the frame bundle of E {\displaystyle E} and using the theory of principal bundles.
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} .
The definition may be phrased for a connection on a vector bundle or principal bundle, with the two perspectives being essentially interchangeable. Here the definition of principal bundles is presented, which is the form that appears in Hitchin's work. [1] [5] [6]
A connection on a vector bundle may be specified similarly to the case for principal bundles above, known as an Ehresmann connection. However vector bundle connections admit a more powerful description in terms of a differential operator. A connection on a vector bundle is a choice of -linear differential operator
The bundle TP/G is called the bundle of principal connections (Kobayashi 1957). A section Γ of dπ:TP/G→TM such that Γ : TM → TP/G is a linear morphism of vector bundles over M, can be identified with a principal connection in P. Conversely, a principal connection as defined above gives rise to such a section Γ of TP/G.
Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundles, and are sometimes called (vector) bundle homomorphisms. A bundle homomorphism from E 1 to E 2 with an inverse which is also a bundle homomorphism (from E 2 to E 1) is called a (vector) bundle isomorphism, and then E 1 and E 2 are said to be ...
If one has a vector bundle E over M, then the metric can be extended to the entire vector bundle, as the bundle metric. One may then define a connection that is compatible with this bundle metric, this is the metric connection. For the special case of E being the tangent bundle TM, the metric connection is called the Riemannian connection.
Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J 1 Y of the jet bundle J 1 Y → Y of Y is an affine bundle morphism over X. In particular, this is an affine connection on the tangent bundle TX of a smooth manifold X. (That is, the ...