Singular Limits of Frame Bundles of Noncollapsing Positively Curved Manifolds

For each dimension $n\ge 2$, we construct noncollapsing sequences of metrics on $S^n$ with a uniform positive lower bound on sectional curvature whose oriented orthonormal frame bundles, equipped with the canonically induced connection metrics, converge to nonmanifold metric spaces homeomorphic to $Σ(S^{n-1}\times\SO(n))$. In even dimensions, real projective quotients give connected full frame bundles with the same limits, disproving the manifold conclusion in Yamaguchi's frame bundle conjecture. In odd dimensions, each component of the full projective frame bundle, with its intrinsic metric, converges to the quotient of the spherical limit by a free involution. The singularities are detected by local homology, and we obtain quantitative equivariant Gromov--Hausdorff estimates. We also compare the frame bundle limits with the intrinsic completions of their regular limiting frame metrics, and present and discuss alternatives to Yamaguchi's original conjecture.