Bochner-type theorems for distributional category

We show that in the presence of a geometric condition such as non-negative Ricci curvature, the distributional category of a manifold may be used to bound invariants, such as the first Betti number and macroscopic dimension, from above. Moreover, à la Bochner, when the bound is an equality, special constraints are imposed on the manifold. We show that the distributional category of a space also bounds the rank of the Gottlieb group, with equality imposing constraints on the fundamental group. These bounds are refined in the setting of cohomologically symplectic manifolds, enabling us to get specific computations for the distributional category and LS-category.

💡 Research Summary

The paper investigates the recently introduced invariant “distributional category” (dcat), a probabilistic analogue of the classical Lusternik–Schnirelmann category (cat), and establishes Bochner‑type inequalities that relate dcat to several fundamental topological and geometric quantities on closed manifolds. After recalling the basic properties of LS‑category, the authors review the definition of dcat (via maps into spaces of probability measures with at most n + 1 support points) and prove that dcat ≤ cat, is invariant under homotopy equivalence, and behaves monotonically under covering maps. They also recall the Gottlieb group G(X) and the result that the rank of any free abelian subgroup of G(X) is bounded above by cat(X).

The central geometric hypothesis is non‑negative Ricci curvature. By invoking the Cheeger–Gromoll splitting theorem, any such manifold M with infinite fundamental group admits a finite cover M′ diffeomorphic to a product T^r × W where W is simply connected. Using the transfer in cohomology, the authors show that the first Betti number satisfies
 b₁(M) ≤ dcat(M) − cup(W),
where cup(W) denotes the rational cup‑length of W. Moreover, equality forces M to be a torus, mirroring the classical Bochner result that b₁(M)=dim M only for flat tori.

When M is additionally c‑symplectic (i.e., it possesses a rational cohomology class ω with ωⁿ = 0), the splitting respects the symplectic form, yielding r = 2k and a decomposition ω = ω₁⊕ω₂ with ω₁∈H²(T^{2k}) and ω₂∈H²(W). In this setting the authors obtain a sharper bound
 b₁(M) ≤ 2·cat(M) − dim M,
and again equality characterises toroidal products.

The paper then turns to flat manifolds, whose fundamental groups are Bieberbach groups. For such manifolds the center Zπ₁(M) coincides with the Gottlieb group, and classical results give rank Zπ₁(M)=b₁(M). Combining this with the inequality rank G(M) ≤ dcat(M) yields b₁(M) ≤ dcat(M). Equality again forces π₁(M)≅ℤⁿ and M≃Tⁿ. Thus the distributional category provides a unified framework that simultaneously recovers Bochner’s original curvature‑Betti bound, its Gottlieb‑group analogue, and new estimates for c‑symplectic manifolds.

The authors illustrate the sharpness of their results with explicit examples: products T^{2k}×N where N is a simply‑connected symplectic manifold, flat Kähler manifolds, and Calabi–Yau manifolds with infinite fundamental group. They also discuss how dcat can improve earlier estimates that only involved LS‑category, especially in cases where cat and dcat differ (e.g., real projective spaces).

Finally, the paper outlines several directions for future work, such as extending the Bochner‑type inequalities to manifolds with lower Ricci bounds, exploring higher Gottlieb groups Gₙ(X) in the dcat context, and investigating potential applications to motion‑planning complexity where dcat naturally arises. In summary, the work establishes distributional category as a powerful tool linking curvature, symplectic structure, and algebraic invariants, and opens a new avenue for Bochner‑type rigidity phenomena in modern topology.