Constant-curvature metric
This article defines a property that makes sense for a Riemannian metric over a differential manifold
This is the property of the following curvature being constant: sectional curvature
Definition
Symbol-free definition
A Riemannian metric on a differential manifold is termed a constant-curvature metric if, for any section of the manifold, the sectional curvature is constant at all points, and moreover, this constant value is the same for all sections.
Equivalently, a Riemannian metric on a differential manifold is termed a constant-curvature metric if it satisfies the following, termed the axiom of free mobility, namely: given any two points in the manifold, and any orthonormal bases for the tangent spaces at the two points, there are neighbourhoods of the two points and a Riemannian isometry from one to the other, that maps one orthonormal basis to the other.
Relation with other properties
Stronger properties
Weaker properties
For manifolds of dimension upto three, constant-curvature metrics are the same as Einstein metrics.