On manifolds with almost non-negative Ricci curvature and integrally-positive k th -scalar curvature

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 ${\pi }_1\left(M\right)$ . 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 $k=n=\text{dim}\left(M\right)$ , 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.