Knowlpedia

Definition

Dual connection

The unique connection on a dual vector bundle for which differentiation obeys the natural evaluation pairing.

Let K∈{R,C}\mathbb K\in\{\mathbb R,\mathbb C\}, and let E→ME\to M be a smooth over K\mathbb K with a ∇\nabla, and let E∗→ME^*\to M be its . The dual connection ∇∗\nabla^* is the unique connection on E∗E^* satisfying

X ⁣(λ(s))=(∇X∗λ)(s)+λ ⁣(∇Xs)X\!\left(\lambda(s)\right) = \left(\nabla_X^*\lambda\right)(s)+\lambda\!\left(\nabla_Xs\right)

for every XX, ss of EE, and section λ\lambda of E∗E^*. Equivalently,

(∇X∗λ)(s)=X ⁣(λ(s))−λ ⁣(∇Xs).\left(\nabla_X^*\lambda\right)(s) =X\!\left(\lambda(s)\right)-\lambda\!\left(\nabla_Xs\right).

The defining identity says exactly that the natural evaluation pairing E∗⊗E→M×KE^*\otimes E\to M\times\mathbb K is parallel.

Local expression

If e1,…,ere_1,\ldots,e_r is a local frame, e1,…,ere^1,\ldots,e^r its dual frame, and

∇ej=∑iAij⊗ei,\nabla e_j=\sum_i A^i{}_j\otimes e_i,

then

∇∗ei=−∑jAij⊗ej.\nabla^*e^i=-\sum_j A^i{}_j\otimes e^j.

Thus the connection matrix on the dual frame is −AT-A^{\mathsf T}. The minus sign is forced by differentiating ei(ej)=δjie^i(e_j)=\delta^i_j.

Curvature

Using the convention R∇(X,Y)=[∇X,∇Y]−∇[X,Y]R^\nabla(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}, the dual curvature satisfies

(R∇∗(X,Y)λ)(s)=−λ ⁣(R∇(X,Y)s).\left(R^{\nabla^*}(X,Y)\lambda\right)(s) =-\lambda\!\left(R^\nabla(X,Y)s\right).

Hence the curvature matrix of the dual connection is the negative transpose of the original curvature matrix.

Conventions and scope
References
  1. Loring W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer, 2017. DOI record. Relevant: induced connections on dual and tensor bundles.
  2. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter II, covariant differentiation on associated tensor bundles.