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
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.
* Gromov, M.: "Filling Riemannian manifolds", ''Journal of Differential Geometry
'' 18 (1983), 1–147.