A great circle divides the sphere in two equal hemispheres
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle equipped with its great-circle distance. It is the circle equipped with its ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or the ''extrinsic'' metric obtained by restriction of the ''intrinsic'' metric on the sphere, as opposed to the ''extrinsic'' metric obtained by restriction of the Euclidean metric to the unit circle in the plane. Thus, the distance between a pair of points is defined to be the length of the shorter of the two arcs into which the circle is partitioned by the two points.
It is named after German mathematician Bernhard Riemann.

** Properties **

The diameter of the Riemannian circle is π, in contrast with the usual value of 2 for the Euclidean diameter of the unit circle.
The inclusion of the Riemannian circle as the equator (or any great circle) of the 2-sphere of constant Gaussian curvature +1, is an isometric imbedding in the sense of metric spaces (there is no isometric imbedding of the Riemannian circle in Hilbert space in this sense).

** Gromov's filling conjecture **

A long-standing open problem, posed by Mikhail Gromov, concerns the calculation of the filling area of the Riemannian circle. The filling area is conjectured to be 2, a value attained by the hemisphere of constant Gaussian curvature +1.

** References **

* Gromov, M.: "Filling Riemannian manifolds", ''Journal of Differential Geometry'' 18 (1983), 1–147.
{{Bernhard Riemann
Category:Riemannian geometry
Category:Circles
Category:Metric geometry
Category:Bernhard Riemann