Knowlpedia

Definition

Dual connection

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

Definition

Let E→ME\to M be a smooth 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, section 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×RE^*\otimes E\to M\times\mathbb R 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. This functorial construction is treated in Tu, Chapter 12.

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.