Dynamical Geometry of Principal Bundle Constrained Systems: Compatible Pairs, Contact Degeneration, and Adapted Metrics

Abstract

We study invariant codimension-one constraints on principal bundles through compatible pairs: a constraint distribution and a nonzero coadjoint field parallel for a principal connection. Pairing the field with the connection gives an invariant one-form whose Levi form separates a horizontal curvature contribution from a vertical coadjoint-orbit contribution. This decomposition yields criteria for integrability, contactness, and characteristic reduction; holonomy and stabilizer reductions describe global existence. For evolving compatible pairs, we identify the mixed-curvature obstruction to compatibility and prove that connection transport preserves the Levi geometry. For initially contact data over a closed base of dimension $2n$, we establish matching bounds for the quadratic $H^{n+1}$ cost of contact degeneration: making the paired curvature vanish on a Darboux ball of radius $r$ in time $T$ costs an amount comparable to $[T\log (R_{\ast }/r){]}^{-1}$. The constructed paths remain contact before $T$, preserve the curvature class, and force the $L^{\infty }$ norm of every transporting velocity gradient to grow at least as $1/[2(T-t)]$ on the collapsing region in Darboux coordinates. On a closed three-manifold, a contact form preserved by a locally free circle action admits an invariant adapted metric with any prescribed positive curl eigenvalue and unit circle generator. We parametrize all such metrics and prove that their space is contractible. Along the circle-bundle degeneration paths, every continuous tensor limit of normalized adapted metrics is degenerate above the collapsing region.

17 pages. Substantially rewritten: corrected the geometric and dynamical formulation of compatible pairs; removed earlier claims on Atiyah splittings, variational derivations, and Spencer cohomology; added quantitative contact-degeneration estimates and invariant adapted metrics. Title and abstract updated