singlecurvature

Single curvature is a term used in differential geometry to describe a surface in which the curvature is the same in all directions at every point. More precisely, a surface S in Euclidean 3-space has single curvature if its two principal curvatures k1 and k2 satisfy k1(p) = k2(p) for all points p on S. Equivalently, the shape operator at each point is a scalar multiple of the identity, meaning the normal curvature is independent of the direction of departure in the tangent plane.

Consequences of this condition include that the normal curvature in any tangent direction is the same, the Gaussian curvature K equals k^2 (where k = k1 = k2), and the mean curvature H equals k. Surfaces with single curvature are also described as totally umbilic.

In Euclidean 3-space, the classification of connected single-curvature surfaces is rigid: a connected surface with single curvature is either a plane (k = 0) or a sphere of some radius R (k = 1/R). These are the complete, example-filling cases. More generally, in spaces of constant curvature, totally umbilic submanifolds are the analogues of planes and spheres: geodesic planes and geodesic spheres.

Examples and non-examples help illustrate the notion. The plane and the sphere are classic single-curvature surfaces. A cylinder has principal curvatures 0 and 1/R, so it does not have single curvature. A cone likewise fails to have equal principal curvatures at most points.

Notes: the term single curvature is often described in modern language as totally umbilic. The concept plays a role in characterizing surfaces whose curvature is locally isotropic in tangent directions.