Sphere Rigidity from Scalar Curvature and Injectivity Radius

Definitions and mathematical statement

Let $n$ denote the dimension; the supplied record does not specify its admissible range. For a smooth Riemannian manifold $(M,g)$, closed means compact and without boundary. Write $\operatorname{scal}_g(p)$ for its scalar curvature at $p$, and define \[ \operatorname{inj}(M,g)=\inf_{p\in M}\operatorname{inj}_g(p), \] where $\operatorname{inj}_g(p)$ is the supremum of the radii $r>0$ for which the exponential map $\exp_p$ restricts to a diffeomorphism from the open radius-$r$ ball in $T_pM$ onto its image. Let $S^n=\{x\in{\mathbb{R}}^{n+1}:\lVert x\rVert=1\}$ carry its standard smooth structure.

The supplied question asks whether there exists $\varepsilon(n)>0$, depending only on $n$, such that every closed smooth $n$-dimensional Riemannian manifold $(M,g)$ satisfies \[ \left[\operatorname{scal}_g(p)\ge n(n-1)\ \text{for every }p\in M \quad\text{and}\quad \operatorname{inj}(M,g)\ge\pi-\varepsilon(n)\right] \ \Longrightarrow\ M\cong_{\mathrm{diff}}S^n. \] The injectivity-radius inequality includes equality. No additional curvature, volume, or derivative bounds are imposed in this formulation. The source does not explicitly require connectedness or specify a lower dimension bound; these are unresolved qualifications requiring review before the intended scope of the assertion can be fixed.

Short English statement

Does a scalar-curvature lower bound of $n(n-1)$, together with injectivity radius sufficiently close to $\pi$ from below, force a closed $n$-manifold to be diffeomorphic to the standard sphere, with the permitted deficit depending only on $n$? The intended dimension range and connectedness assumption remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003548$, titled Sphere Rigidity from Scalar Curvature and Injectivity Radius; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record supplies proposed year 2017 but names no proposer.
• Supplied report citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2018), Oberwolfach Reports, pp. 2223--2298, DOI 10.4171/owr/2017/36. https://doi.org/10.4171/owr/2017/36. The supplied background describes this collection as drawn from workshop reports and their open-problem discussions.
• Supplied literature reference: Tuschmann--Wiemeler, differentiable stability. https://arxiv.org/abs/1408.3006.
• Supplied literature reference: Richard, strengthened injectivity bounds. https://arxiv.org/abs/2404.09573.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially_solved and reports differentiable sphere stability under additional curvature, volume, or derivative bounds, together with strengthened injectivity estimates in selected topologies. It describes the bare scalar-curvature and near-$\pi$ injectivity-radius question as unresolved. This is supplied dated curation, not independently verified; its time-sensitive status and cited claims require independent checking.