Scalar Curvature Question [?18]: Let $X_{\mathrm{fl}}=\mathbb{R}^n/\Gamma$ be a complete flat manifold whose group $\Gamma$ acts by parallel tr

Definitions and mathematical statement

Let $X_{\mathrm{fl}}=\mathbb R^n/\Gamma$ be a complete flat manifold, where $\Gamma$ is a discrete group acting by translations. Let $(X,g)$ be a complete Riemannian $n$-manifold with $\operatorname{Sc}_g\geq0$. Isometric at infinity is understood to mean that there are compact sets $K\subset X$ and $K_{\mathrm{fl}}\subset X_{\mathrm{fl}}$ and an exact Riemannian isometry \[ X\setminus K\ \cong\ X_{\mathrm{fl}}\setminus K_{\mathrm{fl}}. \] Must $X$ be flat, meaning its full Riemann curvature tensor vanishes identically?

The intended noncompact setting requires clarification: a full-rank translation quotient is a compact flat torus, and the complements-of-compact-sets condition can then become empty and impose no actual end geometry. The source does not explicitly exclude this case. Translation groups also exclude rotational-holonomy flat manifolds, so the summary's references to general flat manifolds and Klein bottles concern a broader class than the actual hypothesis.

The exact exterior isometry is retained; it is not weakened to asymptotic agreement of metrics. The supplied summary describes related nonnegative-scalar-curvature rigidity and splitting results but gives no precise hypotheses or verified complete resolution of this statement.

Short English statement

If a complete nonnegative-scalar-curvature manifold agrees exactly outside compact sets with a translation quotient of Euclidean space, must it be flat? Clarify noncompactness and retain the translation-only hypothesis.

Sources

• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?18], occurrence 18, PDF page 24, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; accessed 2026-07-29, PDF-text-normalized extraction. The provenance describes the public author PDF as incomplete.
• The supplied research summary mentions Gromov--Lawson, Schoen--Yau, D. Li, and Cai--Galloway as related rigidity/splitting work, without full citations or the precise hypotheses applying here. Its reference to the flat torus theorem is likewise not a supplied proof of the scalar-curvature assertion.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700018$, source code catalogue entry, Scalar Curvature Question [?18]. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.