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

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...

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...

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...

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...

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

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...

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...

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

terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle E → X {\displaystyle E\to X} written...

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...

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...

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

In mathematics, a Higgs bundle is a pair ( E , φ ) {\displaystyle (E,\varphi )} consisting of a holomorphic vector bundle E and a Higgs field φ {\displaystyle...

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

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

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

mathematics, an adjoint bundle is a vector bundle naturally associated with any smooth principal bundle. The fibers of the adjoint bundle carry a Lie algebra...

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...

A_{\Sigma }} . Given the composite bundle Y {\displaystyle Y} (1), there is the following exact sequence of vector bundles over Y {\displaystyle Y} : 0 →...

vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may...

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...

smooth vector bundle. Let E be a smooth vector bundle over a smooth manifold M, and let g be a smooth symmetric bilinear form on E. The Clifford bundle of...

can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric. If M is...

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

Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi...