Connection on a Hom bundle

Let $E,F\to M$ be smooth real or complex vector bundles over the same scalar field, with connections $\nabla^E$ and $\nabla^F$ . The homomorphism bundle $\operatorname{Hom}(E,F)=E^*\otimes F$ is the tensor product of the dual bundle of $E$ with $F$ , with fiber the linear maps $E_x\to F_x$ . The induced connection on $\operatorname{Hom}(E,F)$ is the unique connection satisfying

(\nabla^{\operatorname{Hom}}_X T)(s) = \nabla^F_X(Ts)-T(\nabla^E_Xs)

for every vector field $X$ , section $T$ of $\operatorname{Hom}(E,F)$ , and section $s$ of $E$ .

Tensor description

Under the canonical identification $\operatorname{Hom}(E,F)\cong E^*\otimes F$ , it is the tensor product connection formed from the dual connection on $E^*$ and $\nabla^F$ .

Curvature formula

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

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

the induced curvature is

R^{\operatorname{Hom}}(X,Y)T = R^F(X,Y)\circ T-T\circ R^E(X,Y).

In particular, on $\operatorname{End}(E)$ one has $R^{\operatorname{End}}(X,Y)T=[R^E(X,Y),T]$ . This commutator formula is the reason curvature naturally acts on endomorphism-valued tensors.

Local expression and parallel maps

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

\nabla^{\operatorname{Hom}}T = dT+A^F T-TA^E.

Consequently $T$ is parallel exactly when it intertwines the two covariant derivatives. A parallel bundle isomorphism identifies both connections and conjugates their curvatures. If $F$ is the trivial line bundle with its trivial connection, the construction recovers the dual connection on $E^*$ .

Conventions and scope

The notation $\operatorname{End}(E)$ means $\operatorname{Hom}(E,E)$ ; no metric is needed. A Hermitian metric identifies $E$ conjugate-linearly with $E^*$ (equivalently, identifies $\overline E$ complex-linearly with $E^*$ ), but that identification is additional data and does not alter the defining formula.

References

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.
Arthur L. Besse, Einstein Manifolds, Springer, 1987. DOI record. Relevant: Appendix A, connections and curvature on tensor bundles.