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)

Comments

Both types of geodesic circles are also considered in a more general setting. The first one is generalized to the concept of a distance sphere in a Riemannian manifold. The generalization of the second one appears under the notion of an extrinsic sphere, which is characterized as a totally umbilical submanifold having non-vanishing parallel mean curvature normal [a3].

References

[a1]	K. Nomizu, K. Yano, "On circles and spheres in Riemannian geometry" Math. Ann. , 210 (1974) pp. 163–170
[a2]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a3]	B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)