Curvature-transitive metric
This article defines a property that makes sense for a Riemannian metric over a differential manifold
Definition
Symbol-free definition
A Riemannian metric on a differential manifold] is termed curvature-transitive if given any two points with the same sectional curvature, there is an isometry of the manifold taking one to the other.
In other words, the sectional curvature is a complete invariant of the isometry class of a point.