Recently, a new homotopy invariant of metric spaces, called the distributional category, was defined, which provides a lower bound to the Lusternik–Schnirelmann (LS) category. In this paper, we obtain several sufficient conditions for the distributional category ( \({{\,\mathrm{\textsf{dcat}}\,}}\) ) of a closed manifold to be maximum, i.e., equal to its classical LS-category ( \({{\,\mathrm{\textsf{cat}}\,}}\) ). These give us many new computations of \({{\,\mathrm{\textsf{dcat}}\,}}\) , especially for some essential manifolds and (generalized) connected sums. In the process, we also determine the \({{\,\mathrm{\textsf{dcat}}\,}}\) of closed 3-manifolds having torsion-free fundamental groups and various closed geometrically decomposable 4-manifolds. Finally, we extend some of our results to closed Alexandrov spaces with curvature bounded below and discuss their \({{\,\mathrm{\textsf{cat}}\,}}\) and \({{\,\mathrm{\textsf{dcat}}\,}}\) in dimension 3.
