Global metrizability in dimension two and for singular sprays

Determine whether every locally metrizable regular spray of scalar curvature in dimension two, or every locally metrizable singular spray of scalar curvature defined on a connected conical region in dimension at least two, is globally metrizable under the same topological hypotheses as Theorem 1.

Background

Theorem 1 establishes that a regular spray of scalar curvature with almost everywhere nonzero Ricci curvature on a manifold of dimension at least three is globally metrizable if and only if it is locally metrizable, provided that the manifold has trivial first de Rham cohomology group. The proof relies on the topology of the punctured tangent fibers, which are simply connected in dimensions at least three.

The authors explicitly state that their proof does not apply in dimension two or to singular sprays, because the relevant fiber topology or the behavior of neighborhoods in the conical domain creates an obstruction. They then ask whether examples exist in these settings that are locally metrizable but not globally metrizable. The paper later supplies such examples on manifolds with nontrivial first de Rham cohomology, but does not resolve the question under the stated topological hypotheses.

References

We do not know whether Theorem \ref{Thm01} is true for the case $n=2$, or the case that {\bf G} is a singular spray defined on a connected conical region $\mathcal{C}$ of $ToM$ (with $n\ge 2$) (see the connectivity of $\mathcal{C}$ in Section \ref{pre} and Lemma \ref{lem21} below). So a natural problem is whether there are such sprays which are locally metrizable but not globally metrizable.

— Global Metrizability on Sprays of Scalar Curvature  (2609.18292 - Duanmu et al., 16 Sep 2026) in Introduction, immediately following Theorem 1; see also Remark 3.1