Geodesic circle ($S_r(p)=\{{\exp }_p(rv):|v|=1\}$)

For $r$ smaller than the normal-coordinate radius, the geodesic circle on a Riemannian surface consists of points at distance $r$ from $p$. The exponential map restricted to the radius-$r$ circle in $T_{p}S$ gives its smooth embedded parameterization. Outside the injectivity range a distance sphere need not be smooth.

Fix $p$ on a Riemannian surface. For sufficiently small $r>0$, the geodesic circles of radius $r$ centered at $p$ and at every sufficiently nearby distinct $q$ meet transversally at exactly two points. In a strongly convex normal neighbourhood, write $q={\exp }_p(\epsilon v)$ and $x(\theta )={\exp }_p(re(\theta ))$. First variation givesThe quotient of the bracketed function by $\epsilon$ extends smoothly and has two simple zeros at zero. The implicit function theorem, uniformly over unit $v$, preserves these two zeros and excludes others. The nonzero angular derivatives prove transversality. The distinct-centre hypothesis is essential: coincident circles are not transverse.