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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|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|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 [[#References|[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 [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Blaschke,   "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''2''' , Springer  (1923)</TD></TR></table>

====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 [[#References|[a3]]].

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