CD meets CAT

TL;DR

This paper explores the relationship between curvature bounds in $CD(K,n)$ spaces and Alexandrov spaces, establishing bounds and properties that connect these geometric notions, and showing that certain $CD(K,n)$ spaces are infinitesimally Hilbertian.

Contribution

It demonstrates that $CD(K,n)$ spaces with upper curvature bounds are Alexandrov spaces with specific curvature bounds and are infinitesimally Hilbertian, linking different curvature conditions.

Findings

If a noncollapsed $CD(K,n)$ space has an upper curvature bound, then $K \

Such spaces are Alexandrov spaces with curvature bounded below by $K - ppa(n-2)$.

$CD(K,n)$ spaces with finite $n$ and curvature bounded above are infinitesimally Hilbertian.

Abstract

We show that if a noncollapsed $CD(K,n)$ space $X$ with $n\ge 2$ has curvature bounded above by $\kappa$ in the sense of Alexandrov then $K\le (n-1)\kappa$ and $X$ is an Alexandrov space of curvature bounded below by $K-\kappa (n-2)$. We also show that if a $CD(K,n)$ space $Y$ with finite $n$ has curvature bounded above then it is infinitesimally Hilbertian.