On Fox's \(m\)-dimensional category and theorems of Bochner type (Q664707)

It was proven previously by \textit{J. Oprea} in [Proc. Am. Math. Soc. 130, No. 3, 833--839 (2002; Zbl 0984.53015)] that for a compact smooth Riemannian manifold with non-negative Ricci curvature one has a Bochner type inequality \(b_1(M) \leq \text{{cat}}(M)\). In the article under review the authors prove a stronger inequality \[ b_1(M) \leq \text{{cat}}_1(M) \] for \(M\) as above. Besides, the authors give an upper bound on the rank of the Gottlieb group \[ \text{{rank}}(G_1(X)) \leq \text{{cat}}_1(X), \] where \(X\) is a normal ANR. This interesting paper also suggests Bochner-type theorems for manifolds of almost non-negative curvature.