Steiner spanners on non-constant curvature Riemannian manifolds

Establish whether, for any fixed d-dimensional Riemannian manifold of non-constant curvature, there exist Steiner (1+ε)-spanners for every n-point subset whose edge count is \tilde O_d(ε^{(1−d)/2}·n), matching the near-optimal bounds achieved in constant-curvature spaces.

Background

The paper constructs near-optimal Steiner (1+ε)-spanners in constant-curvature geometries (Euclidean, spherical, and hyperbolic) using quadtree-based methods, shifts, and hyperbolic tilings. These results match Euclidean lower bounds up to logarithmic factors.

Extending such guarantees beyond constant curvature to general Riemannian manifolds would broaden applicability significantly. The authors explicitly pose this as an open problem, seeking similar edge bounds in more general curved spaces.

References

We believe that the following are especially promising open problems for future work: Let $\mathbb{M}$ be a fixed $d$-dimensional Riemannian manifold (of non-constant curvature). Can one construct a Steiner spanner for any set of $n$ points in $\mathbb{M}$ that has size $\widetilde O_d(\varepsilon{(1-d)/2}n)$?

— Near-Optimal Dynamic Steiner Spanners for Constant-Curvature Spaces (2509.01443 - Kisfaludi-Bak et al., 1 Sep 2025) in Section Conclusion