# Riemannian circle

In metric space theory and Riemannian geometry, the **Riemannian circle** (named after Bernhard Riemann) is a great circle equipped with its great-circle distance. In more detail, the term refers to the circle equipped with its *intrinsic* Riemannian metric of a compact 1-dimensional manifold of total length 2π, 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.

## 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 2π, 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.