# 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.

### Other articles related to "riemannian circle, circle, riemannian":

Pu's Inequality - Filling Area Conjecture
... Of all possible fillings of the Riemannian circle of length by a -dimensional disk with the strongly isometric property, the round hemisphere has the least area ... To explain this formulation, we start with the observation that the equatorial circle of the unit -sphere is a Riemannian circle of length ... More precisely, the Riemannian distance function of is induced from the ambient Riemannian distance on the sphere ...
Riemannian Circle - Gromov's Filling Conjecture
... open problem, posed by Mikhail Gromov, concerns the calculation of the filling area of the Riemannian circle ...

### Famous quotes containing the word circle:

They will mark the stone-battlements
And the circle of them
With a bright stain.
They will cast out the dead
A sight for Priam’s queen to lament
And her frightened daughters.
Hilda Doolittle (1886–1961)