An explicit infinite nonsymmetric cohomogeneity-one family

Definitions and mathematical statement

A smooth action of a compact Lie group $G$ on a manifold $M$ has cohomogeneity one when its principal orbits have codimension one, equivalently the orbit space is one dimensional. The source asks for such manifolds with interesting topology, in particular not homeomorphic to a symmetric space, and for their curvature properties. It does not specify dimension, connectedness, compactness, acting group or a unique meaning of interesting. The supplied partial result addresses closed simply connected seven-manifolds and compact Riemannian symmetric spaces.

The aid reports that for every integer $r\geq1$, the Grove--Ziller slope parameters \[ (p_-,p_+)=(4r+1,1-4r) \] give a smooth simply connected cohomogeneity-one manifold $P_{2r}^7$ with effective $SO(4)$ action and two singular orbits, each of codimension two. In the usual interval-orbit-space model this would be encoded by subgroups $H\subset K_-,K_+\subset SO(4)$ with $K_\pm/H\cong S^1$, and assembled as \[ SO(4)\times_{K_-}D^2\ \cup_{SO(4)/H}\ SO(4)\times_{K_+}D^2. \] The principal and singular subgroups, their embeddings, central effectiveness quotient and boundary-gluing map are not supplied. Thus the slope pair is retained as a reference to the specific Grove--Ziller diagram, not asserted sufficient by itself to define an arbitrary unambiguous manifold without that diagram.

The reported topological and curvature conclusions are \[ H^4(P_{2r};{\mathbb{Z}})\cong{\mathbb{Z}}/(2r), \qquad P_{2r}\text{ admits an }SO(4)\text{-invariant metric of strongly nonnegative curvature}. \] For this term, one uses the standard curvature-operator convention: at every point there is a four-form $\omega$ such that $R+\omega$ is positive semidefinite on $\Lambda^2T_pM$. Since a four-form contributes zero on decomposable two-vectors, this implies nonnegative sectional curvature, but is not a claim of positive sectional curvature. Exact curvature conventions and the construction of the invariant metric require the omitted reference.

The varying order $2r$ of the integral cohomology group distinguishes the reported members up to homeomorphism. The aid additionally invokes the Cartan--Wolf classification and integral cohomology to exclude homeomorphism of every $P_{2r}$ to a compact Riemannian symmetric space. Torsion by itself is not declared a universal obstruction to every symmetric space without that classification and the seven-dimensional simply connected scope. The report gives an infinite family meeting one concrete interpretation of the request; it does not classify all cohomogeneity-one manifolds or their possible curvature metrics.

Short English statement

The aid reports an infinite seven-dimensional family with varying fourth-cohomology torsion and invariant strongly nonnegative curvature. Its nonsymmetry claim uses the relevant symmetric-space classification. The full subgroup diagram is missing, so the slope notation is not presented as a self-contained construction.

Sources

• Local UnsolvedMath registry, permanent ID 20003159, source code catalogue entry; aim-topology-notes.json, notes[246].
• AIM, Manifolds with nonnegative sectional curvature, Problem 34: https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. No proposer, original remarks or literature entry is supplied.
• Grove--Ziller slope diagrams and Cartan--Wolf symmetric-space classification are named in the supplied summary without full bibliographic references or subgroup data.
• Machine-generated research attempt 1: partial result, partially solved, rigor labelled "proved," low novelty confidence. Not peer reviewed; independently verify before citation. Registry updated 14 August 2026.