Positive-Ricci Cohomogeneity Manifolds with Exceptional Orbits

Definitions and mathematical statement

For a smooth action of a Lie group $G$ on a smooth manifold $M$, write $G\cdot x=\{a\cdot x:a\in G\}$ for the orbit of $x$, and $G_x=\{a\in G:a\cdot x=x\}$ for its stabilizer. Suppose the action has a principal orbit type: there is a subgroup $H\leq G$ such that the points whose stabilizers are conjugate to $H$ form an open dense subset $M_{\mathrm{pr}}\subseteq M$. Orbits in this subset are principal; all other orbits are non-principal. Set \[ c=\dim M-\dim(G/H),\qquad \mathcal N=\{G\cdot x:x\in M\setminus M_{\mathrm{pr}}\}. \] Here $c$ is the cohomogeneity, using the principal orbit dimension. A Riemannian metric $g$ is $G$-invariant if $a^*g=g$ for every $a\in G$, and has positive Ricci curvature if \[ \operatorname{Ric}_{g,x}(v,v)>0 \quad\text{for every }x\in M\text{ and every }0\ne v\in T_xM. \] The supplied question asks whether there exists such an action and such an invariant metric satisfying \[ \#\mathcal N\geq 3, \qquad c\in\mathbb Z_{\geq0}\setminus\{3,5\}. \] The inequality counts distinct orbits, not points. The Lie group action framework above explicates the usual terminology; it does not resolve the source's unspecified admissible class of actions and manifolds. In particular, the supplied record does not state whether $G$ must be compact, whether the action must be effective or proper, or whether $M$ must be connected, compact, complete, or without boundary. These hypotheses require confirmation before a fully specified existence statement can be fixed. The displayed existential reading also leaves unresolved whether the intended question instead requests examples in every cohomogeneity other than $3$ and $5$. No finiteness or isolation condition on the set of non-principal orbits is stated. The body asks about all non-principal orbits; the title's reference to exceptional orbits does not justify restricting the question to non-principal orbits of principal dimension.

Short English statement

The question asks for a manifold with a group action having at least three distinct non-principal orbits and a group-invariant metric of positive Ricci curvature, with cohomogeneity different from $3$ and $5$. The admissible manifolds and actions, and whether examples are sought in some or every other cohomogeneity, remain unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30001950$, source problem code catalogue entry, title "Positive-Ricci Cohomogeneity Manifolds with Exceptional Orbits." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year $2012$; no individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Manifolds with Lower Curvature Bounds (2012), Oberwolfach Reports, pp. 5--42, DOI 10.4171/owr/2012/01. Supplied source URL and literature link labelled "original OWR report": https://doi.org/10.4171/owr/2012/01.
• Supplied literature link labelled "many isolated singular orbits": https://mural.maynoothuniversity.ie/id/eprint/6970/1/DW-many_isolated.pdf. No authors or further bibliographic details for this link are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports positive-Ricci examples with many non-principal orbits in cohomogeneities $3$ and $5$, while describing the question for other cohomogeneities as remaining open. This is supplied dated curation, not an independently verified literature assessment; the source record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement "exact" and reports a manual review finding the extracted statement self-contained, mathematically coherent, and free of report-layout artifacts. These are supplied provenance claims, not independent verification of the hypotheses or mathematics in this draft.