Circle (Riemannian manifold)

Definition

The circle, as a Riemannian manifold, or as an intrinsic curve, is defined as a simple closed curve (i.e. diffeomorphic to the circle) satisfying the following equivalent properties:

• Its curvature is constant, and its torsion is zero
• It can be embedded as a submanifold of the plane, as a circle in the plane.