On Foxʼs m-dimensional category and theorems of Bochner type

We show that cat m ( X ) = cat ( j m ) , where cat m ( X ) is Foxʼs m-dimensional category, j m : 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.