Bochner-type theorems for distributional category
Ekansh Jauhari, John Oprea
TL;DR
The paper develops Bochner-type relations for distributional category ($ ext{dcat}$) under non-negative Ricci curvature, establishing bounds on $b_1(M)$, macroscopic dimension, and the rank of Gottlieb groups, with equality forcing rigid structures such as tori or flat manifolds. It leverages Cheeger–Gromoll splitting and extensions to $c$-symplectic manifolds to refine existing LS-category results and to derive precise equalities or inequalities for $ ext{dcat}$ and $ ext{cat}$ in both general and special geometric settings. A key mechanism is the use of Hurewicz splittings $X'\\simeq T^k\times Y$ and the corresponding cup-length decompositions, which connect algebraic invariants to geometric decompositions. The work further ties $ ext{dcat}$ to macroscopic dimension, showing that equality with $ ext{dim}_{mc}$ characterizes flatness, and offers constructions demonstrating gaps between Gottlieb rank and $ ext{dcat}$, along with extensions to $c$-symplectic manifolds and their fundamental groups.
Abstract
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.