(a) Let $S \subset \mathbb{R}^{3}$ be an oriented surface and let $\lambda$ be a real number. Given a point $p \in S$ and a vector $v \in T_{p} S$ with unit norm, show that there exist $\varepsilon>0$ and a unique curve $\gamma:(-\varepsilon, \varepsilon) \rightarrow S$ parametrized by arc-length and with constant geodesic curvature $\lambda$ such that $\gamma(0)=p$ and $\dot{\gamma}(0)=v$.

[You may use the theorem on existence and uniqueness of solutions of ordinary differential equations.]

(b) Let $S$ be an oriented surface with positive Gaussian curvature and diffeomorphic to $S^{2}$. Show that two simple closed geodesics in $S$ must intersect. Is it true that two smooth simple closed curves in $S$ with constant geodesic curvature $\lambda \neq 0$ must intersect?