Sphere Theorems with and without Smoothing

TL;DR

This paper establishes sphere theorems for Riemannian and non-collapsed RCD spaces with curvature bounds and mean distance near b1b7, providing new geometric insights.

Contribution

It proves two sphere theorems applicable to both smooth and non-smooth spaces with curvature and distance conditions, extending classical results.

Findings

Sphere theorems for Riemannian manifolds with scalar curvature bounds

Sphere theorems for non-collapsed RCD spaces with mean distance near b1b7

Extension of classical sphere theorems to non-smooth metric measure spaces

Abstract

We show two sphere theorems for the Riemannian manifolds with scalar curvature bounded below and the non-collapsed $\mathrm{RCD}(n-1,n)$ spaces with mean distance close to $\frac{\pi}{2}$.