Knowlpedia

Definition

Connection on a Hom bundle

The connection on a homomorphism bundle induced by connections on its source and target bundles.

Definition

Let E,F→ME,F\to M be smooth real or complex with ∇E\nabla^E and ∇F\nabla^F. The induced connection on Hom⁡(E,F)\operatorname{Hom}(E,F) is the unique connection satisfying

(∇XHom⁡T)(s)=∇XF(Ts)−T(∇XEs)(\nabla^{\operatorname{Hom}}_X T)(s) = \nabla^F_X(Ts)-T(\nabla^E_Xs)

for every XX, section TT of Hom⁡(E,F)\operatorname{Hom}(E,F), and section ss of EE. Under the canonical identification Hom⁡(E,F)≅E∗⊗F\operatorname{Hom}(E,F)\cong E^*\otimes F, it is the formed from the on E∗E^* and ∇F\nabla^F.

Curvature formula

The of the induced connection obeys the following formula. With the convention

R∇(X,Y)=[∇X,∇Y]−∇[X,Y],R^\nabla(X,Y) = [\nabla_X,\nabla_Y]-\nabla_{[X,Y]},

the induced curvature is

RHom⁡(X,Y)T=RF(X,Y)∘T−T∘RE(X,Y).R^{\operatorname{Hom}}(X,Y)T = R^F(X,Y)\circ T-T\circ R^E(X,Y).

In particular, on End⁡(E)\operatorname{End}(E) one has REnd⁡(X,Y)T=[RE(X,Y),T]R^{\operatorname{End}}(X,Y)T=[R^E(X,Y),T]. This commutator formula is the reason curvature naturally acts on endomorphism-valued tensors; it follows from the standard induced-connection construction in Besse, Appendix A.

Local expression and parallel maps

Choose local frames in which ∇E=d+AE\nabla^E=d+A^E and ∇F=d+AF\nabla^F=d+A^F. A homomorphism field is then a matrix-valued function TT, and

∇Hom⁡T=dT+AFT−TAE.\nabla^{\operatorname{Hom}}T = dT+A^F T-TA^E.

Consequently TT is parallel exactly when it intertwines the two covariant derivatives. A parallel identifies both connections and conjugates their curvatures. If FF is the trivial with its trivial connection, the construction recovers the dual connection on E∗E^*.

Conventions and scope

The notation End⁡(E)\operatorname{End}(E) means Hom⁡(E,E)\operatorname{Hom}(E,E); no metric is needed. A can identify E∗E^* with a conjugate-dual bundle, but that identification is additional data and does not alter the defining formula.

References
  1. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter II, induced connections on associated tensor bundles.
  2. Arthur L. Besse, Einstein Manifolds, Springer, 1987. DOI record. Relevant: Appendix A, connections and curvature on tensor bundles.