We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest k eigenvalues of the Ricci tensor. If \((M^n,g)\) is a Riemannian manifold satisfying such curvature bounds for \(k=2\) , then we show that M is contained in a neighbourhood of controlled width of an isometrically embedded 1-dimensional sub-manifold. From this, we deduce several metric and topological consequences: M has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of M is bounded above by 1, and there is precise information on elements of infinite order in \(\pi _1(M)\) . If \((M^n,g)\) is a Riemannian manifold satisfying such bounds for \(k\ge 2\) , then we show that M has at most \((k-1)\) -dimensional behavior at large scales. If \(k=n=\textrm{dim}(M)\) , so that the integral lower bound is on the scalar curvature, assuming in addition that the \((n-2)\) -Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to \(n-2\) . From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.
