Geodesic circles on $2$-sphere

Question

Consider the 2-sphere $S^2$ with a smooth Riemannian metric $g$ and induced distance function $d$, pick a point $p\in S^2$. For small $r>0$, geodesic circles $S(p,r):=\{q\in S^2\mid d(p,q)=r\}$ are simple closed curves. Is this true (or under what asumptions is it true) for all $r>0$ (if $S(r,p)$ is not the empty set or a single point)?

Answer 1

Let us make this question less mathematical. If you think that the sphere is the earth, and your point $p$ is a village in a valley surrounded by several mountains, your set $S(p,r)$ is the set of point at the same distance from your village. If the distance tow the top of two mountains are say $d, d'$, then the set $S(p, r)$ will consist of a least two closed curves : the points at the distance $d-r$ to the top of the first mountain and another set consisting of point on the second mountain at the distance $d'-r$ to the summit. You can make this example more complicated by requiring several mountains, or asking that the distance to the two summits are exactly the same, and looking what happens at the mountain pass...