Collapse of unit horizontal bundles equipped with a metric of Cheeger-Gromoll type

We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse theorem for unit horizontal bundles associated with a sequence of warped Riemannian submersions.

💡 Research Summary

The paper investigates the geometry of unit horizontal bundles that arise from Riemannian submersions, endowing these bundles with a Cheeger‑Gromoll‑type metric. Starting from a Riemannian submersion π : (M,g) → (N,h), the authors consider the horizontal distribution H = (ker π_*)^⊥ and define the unit horizontal bundle UH = {v ∈ H | ‖v‖_g = 1}. On UH they introduce a metric originally inspired by Cheeger and Gromoll’s construction for complete non‑compact manifolds. The metric splits the tangent space of UH into horizontal and vertical parts; the horizontal part inherits the original metric g, while the vertical part is weighted by a factor φ(v) = (1+‖v‖_g²)⁻¹. This weighting makes the metric non‑product and introduces curvature contributions that depend on the size of the horizontal vector.

The first major technical achievement is the derivation of explicit formulas for the Levi‑Civita connection and the Riemann curvature tensor of (UH, ĝ) in terms of the O’Neill tensors A and T associated with the original submersion. The authors show that the horizontal curvature of UH can be expressed as the pull‑back of the base curvature plus correction terms involving A‑A and T‑T interactions, while the vertical curvature is governed entirely by derivatives of the weighting function φ. These formulas lead to sufficient conditions under which UH has non‑negative sectional curvature: essentially the base (N,h) must have non‑negative curvature and the weighting function must vary slowly enough.

In the second part the paper turns to a collapse phenomenon. The authors consider a family of warped submersions π_ε : (M,g_ε) → (N,h) where the metric on M is deformed by a small parameter ε>0: g_ε = g_H ⊕ ε² g_V. This scaling shrinks the vertical directions while leaving the horizontal directions essentially unchanged. The unit horizontal bundles UH_ε associated with each ε inherit the Cheeger‑Gromoll‑type metric. By carefully estimating distances in UH_ε and comparing them with distances in the base (N,h), the authors prove that as ε → 0 the spaces (UH_ε, ĝ_ε) converge to (N,h) in the Gromov‑Hausdorff sense. The proof relies on three pillars: (i) a distance comparison argument that shows any two points in UH_ε can be connected by a curve whose length is arbitrarily close to the length of its projection on N; (ii) preservation of convexity properties under the scaling; and (iii) control of the O’Neill tensors to guarantee that no “hidden” vertical length survives the limit.

The central collapse theorem can be stated informally as: For a sequence of ε‑warped Riemannian submersions, the corresponding unit horizontal bundles collapse to the base manifold as ε tends to zero. This result extends classical collapse theorems—usually formulated for the total space of a submersion—to the more delicate setting of the unit horizontal bundle equipped with a non‑product metric.

To illustrate the abstract theory, the authors present two concrete examples. The first involves a product manifold S¹ × Bⁿ where the S¹ factor is the fiber. By choosing the warping function f_ε = ε, the unit horizontal bundle collapses to the base Bⁿ, demonstrating how the vertical S¹ direction disappears in the limit. The second example takes a base with positive curvature, such as complex projective space CP^k, and a non‑trivial A‑tensor. Even in this more intricate setting, provided the weighting function is sufficiently tame, the collapse theorem still applies.

The paper concludes by emphasizing that the Cheeger‑Gromoll‑type metric provides a flexible tool for probing the fine geometry of horizontal structures in submersions. Potential future directions include extending the analysis to non‑unit horizontal bundles, exploring more general warping functions, and investigating interactions with additional geometric structures such as group actions or foliations.

Overall, the work offers a comprehensive treatment of unit horizontal bundles with a Cheeger‑Gromoll‑type metric, establishes precise curvature formulas, and proves a robust Gromov‑Hausdorff collapse theorem, thereby opening new avenues for research in Riemannian submersion theory and metric geometry.