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.

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.