Orbit stratifications and smooth topology of positively curved six-manifolds with rank-one nonabelian symmetry

Abstract

We study closed simply connected six-manifolds with positive sectional curvature and an effective isometric action of $SO(3)$ or $SU(2)$. Starting from Liu's corrected orbit-space analysis, we refine the stratification and recover smooth topology in several previously unresolved cases. For a boundaryless $SO(3)$ quotient, singular isotropy is connected, the weighted nonprincipal graph belongs to nine necessary families, and every action with only cyclic exceptional isotropy is classified equivariantly. The remaining branching families admit explicit rational intersection and Pontryagin formulas. For a three-ball $SO(3)$ quotient, the uniform-boundary configurations give sphere bundles over $S^4$ and a completely distinguished odd-integer family of non-spin six-manifolds. A doubled-quotient argument excludes the configuration with two boundary fixed points and an interior singular orbit, while the configuration with one fixed point and an exceptional interval has underlying manifold $\mathbb{CP}^3$. For a boundaryless $SU(2)$ quotient, an adjoint-weight matching argument reduces the exceptional three-orbit case to a unique action on the Eschenburg flag and identifies every two-orbit candidate with a weighted action on $S^2\times S^4$. These classifications imply rational ellipticity throughout the stated symmetry class. We also establish restricted curvature obstructions for the two diagonal product actions, without asserting nonexistence for arbitrary invariant metrics. Abstract graph restrictions, smooth realizability, and positive-curvature realizability are kept distinct. Some of the results in this paper are discovered by ChatGPT-6-astra and verified by the authors.