Connection compatible with a reduction - Imported Knowlpedia Content

Definition

Let $H\subseteq G$ be a Lie subgroup, let $i:Q\hookrightarrow P$ be a reduction of structure group of a principal $G$ -bundle, and let $A$ be a principal connection on $P$ . The connection $A$ is compatible with the reduction $Q$ when its horizontal subspace at every $q\in Q$ lies in $T_qQ$ . Equivalently, the restricted connection form $i^*A$ takes values in the Lie algebra $\mathfrak h\subseteq\mathfrak g$ ; then $i^*A$ , regarded as an $\mathfrak h$ -valued form, is a principal $H$ -connection on $Q$ .

Restriction and extension

A compatible $G$ -connection restricts uniquely to an $H$ -connection on $Q$ . Conversely, every principal $H$ -connection on $Q$ extends uniquely to a principal $G$ -connection on the extended bundle $Q\times_HG$ , hence on $P$ after choosing the reduction isomorphism. This correspondence is the connection-level counterpart of extension of structure group; see Kobayashi and Nomizu, Volume I, Chapter II.

Parallel transport and holonomy

Compatibility means that horizontal lifts beginning in $Q$ remain in $Q$ . Therefore parallel transport preserves the reduced subbundle, and the holonomy group computed from a point $q\in Q$ lies in $H$ . Conversely, the holonomy reduction principle constructs a preserved reduction from suitable holonomy containment, subject to its connectedness and reduction hypotheses.

Conventions and scope

References

Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter II, connections on reductions and extension of structure group.
Arthur L. Besse, Einstein Manifolds, Springer, 1987. DOI record. Relevant: Chapter 10, holonomy reductions and preserved geometric structures.