A quantitative nonnegatively curved collapse to a ray

Definitions and mathematical statement

The source's complete request is "Study collapse to a ray." It does not prescribe dimension, curvature normalization, completeness, pointed versus compact convergence, or a classification target. In the nonnegative-sectional-curvature workshop context, the aid supplies an explicit rotational family rather than a general classification.

Fix $n\geq2$. On ${\mathbb{R}}^n$ in polar coordinates, consider \[ g_i=dr^2+f_i(r)^2g_{S^{n-1}},\qquad r\geq0, \] where $g_{S^{n-1}}$ is the unit round metric, $f_i(r)>0$ for $r>0$, and the pole is smooth: $f_i(0)=0$, $f_i'(0)=1$, with the usual smooth odd extension at zero. Suppose \[ \sup_{r\geq0}f_i(r)\leq a_i,\qquad a_i\longrightarrow0. \] The aid reports that the radius map $\rho_i:({\mathbb{R}}^n,g_i)\to[0,\infty)$ is an exact submetry, sending every metric ball onto the corresponding ray ball. On the ambient closed ball around the pole $o$, with its restricted ambient distance, it reports the uniform estimate \[ d_{\mathrm{GH}}\left(\overline B_{g_i}(o,R),[0,R]\right) \leq\frac{\pi a_i}{2}\qquad\text{for every }R>0. \] The bound is independent of $R$, and yields pointed Gromov--Hausdorff convergence to the ray based at zero. The assumption $n\geq2$ ensures connected spherical angular fibers; the same argument is not asserted for ${\mathbb{R}}^1$.

If in addition $f_i''\leq0$ and $0\leq f_i'\leq1$, the reported nonnegative sectional curvature follows from \[ K_{\mathrm{rad}}=-\frac{f_i''}{f_i},\qquad K_{\mathrm{tan}}=\frac{1-(f_i')^2}{f_i^2}, \] with the tangential formula applying to tangential two-planes when $n\geq3$. Curvature extends smoothly across the pole under the pole conditions.

The explicit example is \[ f_\varepsilon(r)=\varepsilon\tanh(r/\varepsilon),\qquad\varepsilon>0. \] It gives complete smooth nonnegatively curved metrics on ${\mathbb{R}}^n$, $\sup f_\varepsilon=\varepsilon$, and the reported volume estimate \[ \operatorname{Vol}_{g_\varepsilon}\overline B(o,R) \leq\operatorname{Vol}(S^{n-1})\,R\varepsilon^{n-1}. \] Thus every fixed-radius ball collapses in volume for $\varepsilon\to0$. The fibers over $r>0$ are spheres and the fiber over zero is a point, identified as the point soul in this example. This endpoint degeneration is not a globally regular sphere bundle over the closed ray.

These hypotheses impose no common upper sectional-curvature bound. In particular, the example's smooth-pole curvature scales like $\varepsilon^{-2}$, so it must not be presented as a normalized two-sided-bounded-curvature collapse. The record gives an existence model and quantitative radius-map control, but no classification of all nonnegatively curved collapses to a ray or every possible endpoint fiber.

Short English statement

The aid gives complete rotational examples collapsing to a ray, with an error bound uniform over ball radii and an explicit volume rate. The endpoint fiber becomes a point. The examples do not have a uniform upper curvature bound and are not a classification of ray collapse.

Sources

• Local UnsolvedMath registry, permanent ID 20003143, source code catalogue entry; aim-topology-notes.json, notes[230].
• M. Kerin's compilation Open Problems in Non-negative Sectional Curvature, from the September 2007 AIM workshop Manifolds with Non-negative Sectional Curvature, Problem 18: https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.
• The bounded-warping criterion and explicit hyperbolic-tangent model are supplied by the research summary without a separate authoritative external citation.
• Machine-generated research attempt 1: partial result, open at the research run, rigor labelled "proved," low novelty confidence; registry partially solved. Not peer reviewed; independently verify before citation. Registry updated 14 August 2026.