Conjugate Connections

Conjugate Connections

Conjugate Connections

Let be a smooth manifold with an affine connection and metric . The conjugate connection relative to is determined by the equation We also say that and are dual with respect to .

Proposition

Given a manifold , the conjugate connection is uniquely determined.

Proof By the definition of conjugate connection, we can drive the coordinate form by Therefore the conjugate connection coefficients are uniquely determined.

Proposition

Conjugation is an involution: .

Proof By the definition of conjugate connection, we have Therefore, for all . Since the metric is non-degenerate, we have .

Lemma

A pair of conjugate connections on a manifold preserves the metric , in the sense that for all , the parallel transport of and conjugate parallel transport of along any curve with doesn’t change the inner product under :

Proof Since and are conjugate, suppose and are corresponding parallel transport vector fields of and . Then we have That is remains unchanged along .

Theorem

Conjugate of a flat connection is flat.

Proof Suppose is flat. By the above lemma, pick , we have holds for any curve . For arbitrary curves with and , we have By flatness of , the parallel transport with respect to is path independent. That is, Thus It follows that , and hence the parallel transport with respect to is also path independent, is flat as well.

Mean Connection

Mean Connection

Suppose manifold is equipped with a pair of conjugate connections , then the mean connection is defined by the following equation:

Proposition

The mean connection is self-conjugate.

Proof

Theorem

The mean connection coincides with the Levi-Civita metric connection if the conjugate connections are torsion-free.

Proof Suppose and are torsion-free, then clearly the mean connection is torsion-free. By the lemma, the mean connection, which a linear combination of and , also preserves the metric. Therefore, the mean connection is the unique Levi-Civita metric connection.

Corollary

The Levi-Civita connection is self-conjugate.