On Foxʼs m-dimensional category and theorems of Bochner type

We show that catm(X)=cat(jm), where catm(X) is Foxʼs m-dimensional category, jm:X→X[m] is the mth Postnikov section of X and cat(X) is the Lusternik–Schnirelmann category of X. This characterization is used to give new “Bochner-type” bounds on the rank of the Gottlieb group and the first Betti number for manifolds of non-negative Ricci curvature. Finally, we apply these methods to obtain Bochner-type theorems for manifolds of almost non-negative sectional curvature.