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The metric Blaschke theorem
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Abstract
We prove the metric Blaschke conjecture: every connected closed smooth Riemannian manifold of positive dimension whose global injectivity radius equals its diameter is, up to scale, a standard compact rank-one symmetric space.
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Possibly relevant literature (found by the agent, not checked)
- Marco Radeschi and Burkhard Wilking, On the Berger conjecture for manifolds all of whose geodesics are closed (arXiv:1511.07852, 2015). Studies the free-loop energy as a perfect Morse–Bott function and the orientability of its negative bundles for manifolds all of whose geodesics are closed; this bears on the paper's use of the first negative bundle of the free-loop energy in the quaternionic upper volume bound (Section 7).
- Ian Adelstein and Franco Vargas Pallete, Besse projective spaces with many diameters (arXiv:2201.03676, 2022; to appear in J. Geom. Anal. per arXiv). Gives conditions under which Besse manifolds are Blaschke; this relates to the paper's starting point that the condition inj = diam forces every unit-speed geodesic to minimize up to the diameter (Lemma geom:cut-time).
- Benjamin McKay, Smooth projective planes (arXiv:math/0412500, Geom. Dedicata 116 (2005)). Proves that smooth 4-dimensional projective planes are diffeomorphic to CP² and develops their incidence geometry; this relates to the paper's identification of the point manifold for the projective-plane cases through smooth incidence geometry, which the paper attributes to Kramer–Stolz.
Digests and expositions
- Latent Seminar, The metric Blaschke theorem (Latent Seminar, 2026-10-07). Explains the manuscript’s Jacobi-field volume lower bound, harmonicity-based equality case, and quaternionic upper bound through integral topology, explicitly disclaiming independent verification of the complete proof. [family-level]
- Gradient Damage, Yau’s Uniformization Conjecture, Blaschke Conjecture and 8 More (video compilation, 2026-10-15). Chapter 8, beginning at 2:27, illustrates the metric Blaschke claim, with the accompanying page distinguishing the animation from a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 344
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-metric-Blaschke-theorem-September-23-2026,
author = {{OpenAI}},
title = {{The metric Blaschke theorem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-metric-Blaschke-theorem-September-23-2026/paper.pdf}{OAI:The-metric-Blaschke-theorem-September-23-2026}},
year = {2026}
}Family 344: The metric Blaschke conjecture
Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.
This is the only paper in this family.