Metric circle

In mathematics, a metric circle is the metric space of arc length on a circle, or equivalently on any rectifiable simple closed curve of bounded length.

Some authors have called metric circles Riemannian circles, especially in connection with the filling area conjecture in Riemannian geometry,[2] but this term has also been used for other concepts.

The same metric space would also be obtained from distances on a hemisphere.

This differs from the boundary of a unit disk, for which opposite points on the unit disk would have distance 2, instead of their distance π on the Riemannian circle.

This difference in internal metrics between the hemisphere and the disk led Mikhael Gromov to pose his filling area conjecture, according to which the unit hemisphere is the minimum-area surface having the Riemannian circle as its boundary.

Arc distances on a great circle are the same as the distance between the same points on a sphere , and on the hemispheres into which the circle divides the sphere.