# On manifolds with almost non-negative Ricci curvature and integrally-positive kth-scalar curvature

## TL;DR

This paper studies the geometric and topological properties of manifolds with specific curvature conditions, revealing constraints on their volume, structure, and topology.

## Contribution

The paper introduces new curvature bounds and derives novel topological and metric consequences for manifolds under these conditions.

## Key findings

- Manifolds with curvature bounds for k=2 are near a 1-dimensional sub-manifold.
- Such manifolds have linear volume growth and at most two ends.
- Topological restrictions include an upper bound on the first Betti number.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$$(M^n,g)$$\end{document}(Mn,g) is a Riemannian manifold satisfying such curvature bounds for \documentclass[12pt]{minimal}
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\begin{document}$$k=2$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\pi _1(M)$$\end{document}π1(M). If \documentclass[12pt]{minimal}
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\begin{document}$$(M^n,g)$$\end{document}(Mn,g) is a Riemannian manifold satisfying such bounds for \documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document}k≥2, then we show that M has at most \documentclass[12pt]{minimal}
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\begin{document}$$(k-1)$$\end{document}(k-1)-dimensional behavior at large scales. If \documentclass[12pt]{minimal}
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\begin{document}$$k=n=\textrm{dim}(M)$$\end{document}k=n=dim(M), so that the integral lower bound is on the scalar curvature, assuming in addition that the \documentclass[12pt]{minimal}
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\begin{document}$$(n-2)$$\end{document}(n-2)-Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to \documentclass[12pt]{minimal}
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\begin{document}$$n-2$$\end{document}n-2. From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.