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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.
It is not always possible for an Ehresmann connection to induce, in a natural way, a connection on an associated bundle. For example, a non-equivariant Ehresmann connection on a bundle of frames of a vector bundle may not induce a connection on the vector bundle. Suppose that E is an associated bundle of P, so that E = P × G F.
These are examples of affine connections. There is also a concept of projective connection, of which the Schwarzian derivative in complex analysis is an instance. More generally, both affine and projective connections are types of Cartan connections. Using principal bundles, a connection can be realized as a Lie algebra-valued differential form.
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.
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.
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} .
Connections on a surface can be defined in a variety of ways. The Riemannian connection or Levi-Civita connection [9] is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of ...
Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called instantons. The Kobayashi–Hitchin correspondence proved by Donaldson, Uhlenbeck and Yau asserts that a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian Yang–Mills connection if and only if it is slope polystable.