Scalar Curvature Question [?66]: Probably, these equalities imply thatP is isometric to a Euclidean rectangular solidbut the approximation/smoo

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a Riemannian domain, mean curvature $H$ of a smooth boundary face is the trace of its second fundamental form with the convention that the boundary of a Euclidean ball has positive mean curvature. Mean convexity means $H\geq0$. Dihedral angles are interior angles between adjacent faces, measured between their tangent spaces along their common codimension-two faces. Let $P=[0,1]^n$ be the $n$-cube equipped with a Riemannian metric $g$, with its usual faces and corners. Write $\operatorname{Sc}_g$ for scalar curvature, the trace of Ricci curvature, $H_g(Q_i)$ for the mean curvature of each codimension-one face, and $\alpha_{ij}(g)$ for the interior dihedral angles along adjoining faces. In Gromov's 2017 formulation, the hypotheses $\operatorname{Sc}_g\geq0$, $H_g(Q_i)\geq0$ and $\alpha_{ij}(g)\leq\pi/2$ yield the equalities \[\operatorname{Sc}_g=0,\qquad H_g(Q_i)=0,\qquad\alpha_{ij}(g)=\pi/2.\] A Euclidean rectangular solid is $\prod_{i=1}^n[0,\ell_i]$ with $\ell_i>0$, equipped with its ordinary Euclidean metric.

Statement recorded in the supplied source.

Do these equalities imply that $(P,g)$ is isometric to a Euclidean rectangular solid? [S2]

Short English statement

Must a Riemannian cube with zero scalar curvature, minimal faces and right dihedral angles be a Euclidean rectangular solid?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6700062 (AMR-066-0062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Misha Gromov, A Dozen Problems, Questions and Conjectures about Positive Scalar Curvature, arXiv:1710.05946v1 (2017), §7, assertion [*] and following rigidity question, printed p. 21. https://arxiv.org/abs/1710.05946v1.