= Close geodesic circles intersect twice
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 $q=\exp_p(\varepsilon v)$ and $x(\theta)=\exp_p(re(\theta))$. First variation gives
$$
\left.\partial_\varepsilon[d(q,x(\theta))^2-r^2]\right|_{\varepsilon=0}
=-2r\langle v,e(\theta)\rangle.
$$
The quotient of the bracketed function by $\varepsilon$ 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.
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