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Choose any connection form ω in P, and let Ω be the associated curvature form; i.e., =, the exterior covariant derivative of ω. If [] is a homogeneous polynomial function of degree k; i.e., () = for any complex number a and x in , then, viewing f as a symmetric multilinear functional on (see the ring of polynomial functions), let
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 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.
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.
affine connection; conformal connection; projective connection; method of moving frames; Cartan's equivalence method; Vierbein, tetrad; Cartan connection applications; Einstein–Cartan theory; connection (vector bundle) connection (principal bundle) Ehresmann connection; curvature. curvature form; holonomy, local holonomy; Chern–Weil ...
The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure ...
The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel.Levi-Civita, [1] along with Gregorio Ricci-Curbastro, used Christoffel's symbols [2] to define the notion of parallel transport and explore the relationship of parallel transport with the curvature, thus developing the modern notion of holonomy.
Let G be a Lie group with Lie algebra, and P → B be a principal G-bundle.Let ω be an Ehresmann connection on P (which is a -valued one-form on P).. Then the curvature form is the -valued 2-form on P defined by