Product of two CAT($kappa$) spaces is CAT($kappa$) for $kappa ge 0$
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite 1 I would like to see a "metric proof" that if two metric spaces $X$ and $Y$ are CAT($kappa$) for some $kappa ge 0$, then so is their product. I would be satisfied to see a proof for $X=Y=S^2$. By "metric proof" I mean one which does not rely on Riemannian geometry, but rather only uses Alexandrov (metric) geometry. I already understand the case where $kappa le 0$ (where in fact the product will only be CAT($textmax(0,kappa)$). metric-geometry share | cite | improve this question asked Sep 6 at 11:04 Delfador Logalmier 11 2 What is $rm CAT[k]$ and their product ? – HK Lee Sep 6 at 11:36 A CAT($kappa$) space is a geodesic space all of whose geodesic triangles of perimeter less than $2D_kappa$ satisfy the CAT$(kappa)$ inequality. Here $D_kappa= +infty$ if $kappa leq 0$ or $fracpikappa$ if $kappa > 0$. One version of the CAT$(kappa$) in...