CAT(k) space

Summary

A CAT($k$) space is a metric space whose geodesic triangles are at least as thin as triangles in a model surface. The model surface has constant curvature $k$. It is the metric version of the condition "sectional curvature at most $k$". The notion applies to spaces with no smooth structure, such as trees, polyhedral complexes and buildings. CAT(0) spaces are central in geometric group theory.