The maximal symmetry-rank conjecture and a sharp product defect formula

Definitions and mathematical statement

Let $(M^n,g)$ be a closed connected simply connected Riemannian manifold of dimension $n\geq2$ with nonnegative sectional curvature: $\sec_g(\sigma)\geq0$ for every tangent two-plane $\sigma$. Its isometry group $\operatorname{Isom}(M,g)$ is a compact Lie group. Define its symmetry rank to be the dimension of a maximal torus in the identity component of this group. Equivalently, it is the largest integer $r$ for which the torus $T^r=(S^1)^r$ acts effectively and isometrically on $(M,g)$. Effective means that no nonidentity group element fixes every point of the manifold.

The proposed bound asks whether every such manifold satisfies \[ \operatorname{rank}\operatorname{Isom}(M,g) \leq\left\lfloor\frac{2n}{3}\right\rfloor. \] The integer floor is simply the integral form of the source inequality $r\leq2n/3$. The metric is arbitrary within the nonnegative-curvature class, and the bound is to depend only on dimension. No assumption that the torus action is free, has a fixed point, or has a specified orbit type is imposed. The problem concerns the maximal dimension of a torus subgroup, not the dimension of the entire isometry group, which is a different invariant. Simple connectivity is a hypothesis on the underlying manifold and cannot be dropped merely because some orbits are connected. A counterexample would consist of a single such metric with an effective isometric torus action exceeding the displayed rank.

Short English statement

Is the symmetry rank of a simply connected, nonnegatively curved closed manifold at most two thirds of its dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003155 (AIM-TOPOLOGY-0243), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Manifolds with nonnegative sectional curvature, item 30. Original workshop question underlying AIM-TOPOLOGY-0243. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.
• S3 Christine Escher and Catherine Searle, Non-negative curvature and torus actions. General conjecture and special cases. https://arxiv.org/abs/1506.08685.