CD meets CAT

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.

We add a new section where we prove a new theorem that if a $CD(K,n)$ space with finite $n$ has curvature bounded above then it is infinitesimally Hilbertian. Using this we remove the infinitesimal Hilbertianness assumption from the main theorem. Minor corrections, additional references