# A softer connectivity principle

## TL;DR

This paper extends classical geometric theorems to the context of ${k}$-th Ricci curvature, providing soft, optimal conditions that hold on open sets of metrics in the $C^{2}$-topology.

## Contribution

It introduces soft, quantitatively optimal extensions of key theorems to ${k}$-th Ricci curvature, broadening their applicability in Riemannian geometry.

## Key findings

- Extended Sphere Theorem to ${k}$-th Ricci curvature
- Generalized Wilking's connectivity principle
- Adapted Frankel's Theorem for ${k}$-th Ricci curvature

## Abstract

We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking's connectivity principle and Frankel's Theorem to the context of ${k}$-th Ricci curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the $C^{2}$-topology.

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Source: https://tomesphere.com/paper/1812.01021