(i) Let $S \subset \mathbb{R}^{3}$ be a regular surface. Define the notions exponential map, geodesic polar coordinates, geodesic circles.

(ii) State and prove Gauss' lemma.

(iii) Let $S$ be a regular surface. For fixed $r>0$, and points $p, q$ in $S$, let $S_{r}(p)$, $S_{r}(q)$ denote the geodesic circles around $p, q$, respectively, of radius $r$. Show the following statement: for each $p \in S$, there exists an $r=r(p)>0$ and a neighborhood $\mathcal{U}_{p}$ containing $p$ such that for all $q \in \mathcal{U}_{p}$, the sets $S_{r}(p)$ and $S_{r}(q)$ are smooth 1-dimensional manifolds which intersect transversally. What is the cardinality $\bmod 2$ of $S_{r}(p) \cap S_{r}(q)$ ?