metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

theory and

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

, the Riemannian circle is a

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

with a characteristic

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

. It is the circle equipped with the ''intrinsic''

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...

of a compact one-dimensional manifold of total

2, or the ''extrinsic'' metric obtained by restriction of the ''intrinsic'' metric to the two-dimensional surface of the sphere, rather than the ''extrinsic'' metric obtained by restriction of the

Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...

to the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

of the two-dimensional

Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

. The distance between a pair of points on the surface of the sphere 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 German(s) may refer to: * Germany (of or related to) ** Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...

Properties

The

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

of the Riemannian circle is , 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

) of the 2-sphere of constant

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

+1, is an isometric imbedding in the sense of metric spaces (there is no isometric imbedding of the Riemannian circle in

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

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.

References

* Gromov, M.: "Filling Riemannian manifolds", ''Journal of

Differential Geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

'' 18 (1983), 1–147. {{Bernhard Riemann Riemannian geometry Circles Metric geometry Bernhard Riemann