Riemannian circle
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In metric space theory and Riemannian geometry, the term Riemannian circle refers to 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.
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[edit] Diameter
The diameter of the Riemannian circle is π, in contrast with the usual value, 2, of the Euclidean diameter of the unit circle.
[edit] 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.
[edit] Isometric imbedding
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).
[edit] References
- Gromov, M.: "Filling Riemannian manifolds", Journal of Differential Geometry 18 (1983), 1–147.
