vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields...

values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an...

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space...

connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead...

basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms...

(Ehresmann) connections on any fiber bundle associated to P {\displaystyle P} via the associated bundle construction. In particular, on any associated vector bundle...

metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will...

framework) Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold Connection (affine bundle) Connection...

bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may...

relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves the (pseudo-)Riemannian...

notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection. Historically, at the turn of the 20th century...

physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be...

Yang–Mills connection (or Hermite–Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler...

with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold...

a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations...

algebra-valued forms. (A connection form is an example of such a form.) Let M be a smooth manifold and E → M be a smooth vector bundle over M. We denote the...

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and...

secondary vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle...

differentiable principal bundle or vector bundle with a connection. Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. Suppose...

bundle I-bundle Natural bundle Principal bundle Projective bundle Pullback bundle Quasifibration Universal bundle Vector bundle Wu–Yang dictionary Seifert...

connection: see Ehresmann connection, connection (principal bundle) or connection (vector bundle). It is one of the numbers that are important in the Einstein...

vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle π : E → B...

frame bundle (principal bundle) of M (or equivalently, a connection on the tangent bundle (vector bundle) of M). A key aspect of the Cartan connection point...

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 → J1Y of the jet bundle J1Y...

Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)...

differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be...

mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection. Let π : E →...

is a manifold. As such, the fiber is a vector space and the tensor bundle is a special kind of vector bundle. Lee, John M. (2012). Introduction to Smooth...

differentiating of vector fields and tensor fields on a manifold, or, more generally, sections of a vector bundle: see Connection (vector bundle). This ultimately...

tangent bundle. More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle. For the...