Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature

Abstract: Let $M_i$ be a sequence of non-collapsed $n$-manifolds with two-sidedly bounded Ricci curvature. We show that the Gromov-Haudorff limit space, $Y$, of the associated sequence of orthonormal frame bundles, $FM_i$, equipped with an almost canonical metric, shares similar properties as a Ricci limit space of non-collapsing sequence i.e., the singular set has codimension $\ge 4$ whose complement contains an open and dense $C<sup>{1,α}$-Riemannian manifold.

• The paper establishes that with a new O(n)-invariant metric, the Gromov-Hausdorff limit of orthonormal frame bundles recovers key features, including a codimension four singular stratification of Ricci limit spaces.
• It shows the quotient of the limit space under O(n) is isometric to the base Ricci limit space, with the regular set forming a dense C¹,α manifold and uniform curvature bounds maintained away from singularities.
• The construction leverages harmonic radius smoothing and O'Neill’s formula to guarantee uniform Ricci bounds, offering a robust framework for analyzing singular fibers and holonomy in geometric analysis.

Gromov-Hausdorff Limits of Orthonormal Frame Bundles for Non-collapsed Manifolds with Bounded Ricci Curvature

Overview and Motivation

The paper "Gromov-Hausdorff limit of orthonormal frame bundles of non-collapsed manifolds with bounded Ricci curvature" (2604.26399) examines equivariant Gromov-Hausdorff limits of orthonormal frame bundles $FM_i$ associated to a sequence of compact Riemannian $n$-manifolds $(M_i, g_i)$ with two-sided Ricci bounds, bounded diameter, and non-collapsed volume. This inquiry extends classical results on Ricci limit spaces by Cheeger-Colding and Cheeger-Naber to the setting of frame bundles, using a novel construction of a canonical $O(n)$-invariant metric, rather than the typical lifting metric. The principal aim is to analyze the singularity structure, stratification, and regularity properties of limit spaces $Y$ in relation to their base spaces $X$ under these geometric bounds.

Main Results and Theorems

The central theorem asserts that for such a sequence $(M_i, g_i)$, equipped with the new $O(n)$-invariant metric $\tilde{g}_i$ on $FM_i$, the equivariant Gromov-Hausdorff limit $n$0 shares key structural features with the Ricci limit space $n$1:

• The quotient $n$2 is isometric to $n$3.
• The singular set $n$4 of $n$5 has codimension at least $n$6, mirroring Cheeger-Naber's codimension four theorem for Ricci limit spaces, and is contained in the preimage $n$7.
• The regular set $n$8 is open, dense, and forms a $n$9-Riemannian manifold.
• The $(M_i, g_i)$0 metric is constructed such that over regions sufficiently away from $(M_i, g_i)$1, the Ricci curvature of the frame bundle is uniformly bounded.

A further significant claim is that the isotropy group $(M_i, g_i)$2 at $(M_i, g_i)$3 is determined by an infinitesimal holonomy group $(M_i, g_i)$4 passed from $(M_i, g_i)$5 at $(M_i, g_i)$6. Points in $(M_i, g_i)$7 with non-trivial isotropy groups map to singular points in $(M_i, g_i)$8, and infinite isotropy groups indicate singularity in $(M_i, g_i)$9 itself.

Importantly, it is carefully proven that the frame bundle limit $O(n)$0 may not itself be a Ricci limit space and may exhibit non-Euclidean tangent cones at singular points, as illustrated by the Eguchi-Hanson example. The regularity of $O(n)$1 is shown to be substantially more robust than under the canonical lifting metric, which depends on higher curvature derivatives and fails to maintain uniform Ricci bounds.

Construction of the New Metric

To circumvent the technical limitations of canonical metrics, the authors employ a smoothing approach: each $O(n)$2 is approximated by a global metric $O(n)$3 with controlled $O(n)$4-harmonic radius, following methods of Petersen-Wei-Ye and Cheeger-Tian. The metric $O(n)$5 is then defined by lifting $O(n)$6 along the horizontal distribution induced via the Levi-Civita connection of $O(n)$7. This guarantees that the induced metric on $O(n)$8 has regularity properties and curvature bounds tied directly to those in the base manifold.

Ricci bounds for the regular part of $O(n)$9 are established using O'Neill's formula, with precise control furnished by the harmonic radius away from $Y$0. Key technical results show that the Ricci curvature of $Y$1 is uniformly bounded wherever the harmonic radius is uniformly positive, and thus the regular set $Y$2 admits a $Y$3-structure.

Stratification, Holonomy, and Singular Fibers

The singular set $Y$4 is described through infinitesimal holonomy groups using Gromov-Hausdorff convergence theory for vector bundles with Sasaki-type metrics (Solórzano). At singular points, the fibers may have reduced symmetry, leading to stratification and rectifiability analogous to Ricci limit spaces. The codimension four stratification is inherited from Cheeger-Naber's theory; tangent cones at most singular points are products of Euclidean spaces with cones over spherical quotients, but may, in general, fail to be Euclidean.

The paper rigorously analyzes the relationship between singular fibers in $Y$5 and singular points in $Y$6, showing a necessary and sufficient condition for the consistency of $Y$7 and the completion of the orthonormal frame bundle over $Y$8 in terms of the equality (up to conjugacy) of infinitesimal holonomy groups.

Examples are provided to demonstrate the failure of the main results when only lower Ricci bounds are assumed, and to highlight cases where the orthonormal frame bundle resolves certain singularities in the base space.

Implications and Directions

The results bear substantial theoretical significance for geometric analysis, particularly in the study of convergence and degeneration of manifolds with bounded Ricci curvature. The refined construction of lifting metrics in the frame bundles enables better preservation of curvature bounds and regularity properties in limit spaces. This provides technical tools applicable to collapsing sequences with local volume bounded covering geometry, and improves upon the singular fibration theorem of Fukaya in the bounded sectional curvature context.

The deep interplay between fiber geometry, holonomy, and singularity stratification informs future investigations on the topology and metric geometry of spaces emerging as limits under various curvature bounds. The explicit codimension estimates and $Y$9-manifold structures have direct relevance for moduli theory and metric measure geometry.

On a practical level, these insights may influence algorithms in geometric data analysis—where manifold learning and fiber bundle structures are paramount—and inform the design of geometric flows and regularization schemes in computational geometry.

Conclusion

This paper provides a comprehensive extension of the structure theory for Ricci limit spaces to the context of orthonormal frame bundles, leveraging a new metric construction to overcome obstructions posed by classical liftings. The codimension stratification, regularity of the limit space, and detailed holonomy analysis shed light on subtle geometric properties emerging from bounded Ricci curvature assumptions. The work establishes a rigorous foundation for future studies on singularities in Riemannian convergence and for practical applications in geometric analysis across mathematics and related fields.