Geodesic circle

The set of points on a metric two-dimensional manifold whose distance from a fixed point $O$ is a constant $r$. A special case is a circle in the Euclidean plane.

If $r$ is small, a geodesic circle on a regular surface and, in general, in a two-dimensional Riemannian space is a simple closed curve (not necessarily of a constant geodesic curvature); each one of its points may be connected with $O$ by a unique shortest line (the radius or radial geodesic), forming a right angle with the geodesic circle; a geodesic circle bounds a convex region. If $r \rightarrow 0$, the length $l$ of a geodesic circle is connected with the Gaussian curvature $K$ at the point $O$ by the relation

$$\frac{2 \pi r - l }{r ^ {3} } \rightarrow \ { \frac \pi {3} } K.$$

If $r$ is large, more than one radial geodesic may lead to the same point on the geodesic circle, the circle may bound a non-convex region and may consist of several components. A geodesic circle is frequently employed in studies in global geometry. The properties of geodesic circles on general convex surfaces and in manifolds with an irregular metric were studied in [1].

A geodesic circle in the sense of Darboux is a closed curve of constant geodesic curvature. It is a stationary curve for the isoperimetric problem. It coincides with an ordinary geodesic circle on surfaces of constant curvature [2].

References

 [1] Yu.D. Bugaro, M.B. Stratilatova, "Circumferences on a surface" Proc. Steklov Inst. Math. , 76 (1965) pp. 109–141 Trudy Mat. Inst. Steklov. , 76 (1965) pp. 88–114 [2] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , 2 , Springer (1923)