22 problems

Farrell–Zdravkovska conjecture. Every flat manifold bounds geometrically; equivalently, every flat manifold occurs as the cusp cross-section of a one-cusped hyperbolic manifold.

Finite-blocking flatness conjecture. If $(M,g)$) has finite blocking, then $g$ is a flat metric.

Let $M=\mathbf{R}^n/\Gamma$ be a closed flat $n$-manifold, where $\Gamma$ is a discrete, torsion-free, cocompact subgroup of $\operatorname{Isom}(\mathbf{R}^n)$. A cusp cross-secti…

Multiplicity-free holonomy conjecture. For every finite group $H$, there exists a Bieberbach group $\Gamma$ with $\mathbb{Q}$-multiplicity-free holonomy representation…

Orbit lattice conjecture for platycosms. All 10 platycosms have the orbit lattice property.

Consider the infinite Möbius strip, namely the surface between two horizontal lines in the plane, where the lines are identified with a flip. Infinite Möbius-strip conjecture. The…

Let a perimeter-minimizing double bubble be a double bubble enclosing prescribed areas with least perimeter on a flat Klein bottle. Klein bottle conjecture. Perimeter-minimizing do…

Let $(M,g)$ be a flat manifold and let $\Delta_g+V$ be a Schrödinger operator on $(M,g)$. Generic-potential bounded-eigenfunction problem. For generic $V$, are the eigenfunctions u…

Let $g$ be a Riemannian metric on the torus $\mathbb{R}^n/\mathbb{Z}^n$, and suppose that its geodesic flow is toric integrable. Toric integrability conjecture. Then $g$ is flat. T…

Let $M$ be a closed $n$-dimensional manifold with almost nonnegative Ricci curvature, meaning that its diameter and Ricci curvature satisfy … Here $b_1(M)$ denotes the first Betti…

Pontryagin-class non-vanishing conjecture. For the orientable flat manifolds $\widehat{\mathsf{LS}}_n$, the mod $2$ reduction of $p_i(\widehat{\mathsf{LS}}_n)$ is non-zero whenever…

Marcus conjecture. The flat affine structure on $M$ is complete if and only if its flat connection preserves a volume form.

Let $G$ be a non-trivial finite group and let … be a representation. Write $\rho^{\mathbb{R}}$ for the representation obtained by extending scalars to $\mathbb{R}$, and let…

Let $G1$ through $G6$ denote the compact, orientable flat three-manifolds described in the source, and consider their spin structures; among these manifolds, $T^3$ is the three-dim…

Let $H>0$ and $g\in\mathbb{N}\cup\{0\}$. Consider the moduli space of non-congruent, connected, closed $H$-surfaces of genus at most $g$ in a fixed flat $3$-torus. Finiteness conje…

Let $n\geq 3$, and let $(M^n,\bar{g})$ be a connected closed flat Riemannian manifold. A metric has pointwise positive Q-curvature if its Q-curvature is positive at every point. Ri…

Let $Q$ be a closed flat manifold. A complete finite-volume noncompact real hyperbolic manifold has cusps diffeomorphic to $Q\times [b,\infty)$; $Q$ bounds geometrically a hyperbol…

Nonexistence conjecture. No Hantzsche–Wendt manifold has a $\operatorname{Spin}^{\mathbb{C}}$-structure.

Let $T^2$ be a Riemannian $2$-torus without conjugate points. Morse–Hedlund conjecture. The torus $T^2$ is flat. This conjecture is a classical rigidity statement for Riemannian su…

A closed flat manifold is a closed manifold with a flat Riemannian metric. A closed flat manifold bounds geometrically a hyperbolic manifold when it occurs as the cross-section of…

Theorem A conjecture. The conclusion of Theorem A should hold for any finite group.

Let $(M,g)$ be a Riemannian torus, meaning a manifold diffeomorphic to a torus equipped with a Riemannian metric. A finite set $B\subset M\setminus\{x,y\}$ is a blocking set for po…