A moduli correction and fixed-volume factor obstruction

Definitions and mathematical statement

The source's Conjecture $7.2.3$, attributed to Spatzier, says that if $(M,g)$ is a closed higher-rank symmetric space of noncompact type, every metric on $M$ with no conjugate points is a homothetic rescaling of $g$. For a closed manifold this is read as locally symmetric, with noncompact-type symmetric universal cover. The printed statement neither passes to metrics modulo diffeomorphism nor restricts irreducibility. Both issues matter.

First, $\varphi^*g$ has no conjugate points for every diffeomorphism $\varphi:M\to M$, without needing to equal $cg$ as a tensor in the original coordinates. A metric-rigidity conclusion should therefore be of the form $h=\varphi^*(cg)$, equivalently isometry up to scale, rather than literal tensor equality.

Second, the aid gives a reducible higher-rank counterexample that survives that correction. Let $(S_1,g_1),(S_2,g_2)$ be closed hyperbolic surfaces of curvature $-1$, and set \[ M=S_1\times S_2,\qquad g=g_1\oplus g_2,\qquad h_s=e^s g_1\oplus e^{-s}g_2\quad(s\in{\mathbb{R}}). \] These are nonpositively curved, hence have no conjugate points, and \[ \operatorname{Vol}(M,h_s)=\operatorname{Vol}(M,g). \] For $s\ne0$, the two factor curvature scales differ, so $h_s$ is not isometric to any global homothety of $g$, even after a diffeomorphism. The aid gives the exact entropy profile \[ h_{\mathrm{vol}}(h_s)^2=e^{-s}+e^s=2\cosh s, \qquad h_{\mathrm{top}}(h_s)=h_{\mathrm{vol}}(h_s). \] Here volume entropy is the exponential volume-growth rate of balls in the universal cover, and topological entropy is that of the unit-speed geodesic flow. The family has fixed total volume but variable factor scales and entropy.

A viable irreducible version asks: if $(M,g)$ is a closed irreducible locally symmetric manifold of noncompact type and rank at least two, does every Riemannian $h$ without conjugate points satisfy $h=\varphi^*(cg)$ for some $c>0$ and diffeomorphism $\varphi$? Alternatively a reducible formulation must allow factorwise scales. The supplied aid reports the irreducible no-conjugate-points version open in primary literature checked through 2026, and attributes a rigidity result under the stronger no-focal-points hypothesis to Watkins, without a full theorem statement or citation. The counterexample does not refute this corrected irreducible question.

The merged tail starts Section $7.3$: for a closed surface $S$ of genus at least two and a no-conjugate-points metric on $S\times S^1$, is it a direct product? This heading and setup are preserved as a separate unfinished question. Whether direct product means the given coordinates, an isometric global product, a finite-cover product, or splitting only of the universal cover is not specified in the excerpt, and the factor-rescaling example above does not answer it.

Short English statement

The printed rigidity claim needs a diffeomorphism quotient and an irreducibility condition or factorwise scales. A product of hyperbolic surfaces has explicit fixed-volume, no-conjugate metrics with unequal factor scales and entropy squared $2\cosh s$. The corrected irreducible conjecture and the appended surface-times-circle question remain separate.

Sources

• Local UnsolvedMath registry, permanent ID $20001907$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[244]. Source item $7.2$, visible conjecture $7.2.3$, attributed to Spatzier.
• AIM workshop "Geodesics," Burns--Matveev report "Open problems and questions about geodesics": https://aimath.org/pastworkshops/geodesicsproblems.pdf. The extraction also includes the heading and introductory setup of Section $7.3$. No PDF was opened in this drafting session.
• The aid attributes the no-focal-points rigidity case to Watkins and qualifies the corrected irreducible no-conjugate-points literature assessment as checked through 2026. No authoritative full citation or precise theorem hypotheses are supplied. Original remarks and literature fields are empty; proposer/year registry fields are null despite the Spatzier attribution.
• Supplied research attempt $1$, result "counterexample," run status "refuted," rigor label "proved," novelty confidence "low." Registry status "solved," classification SOLVED-IN-LITERATURE, updated 14 August 2026. These labels concern the printed unrestricted formulation, not the corrected conjecture.