Let $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define what it means for a curve $\gamma: I \rightarrow S$ to be a geodesic, where $I=(a, b)$ and $-\infty \leqslant a<b \leqslant \infty$.

(b) A geodesic $\gamma: I \rightarrow S$ is said to be maximal if any geodesic $\widetilde{\gamma}: \tilde{I} \rightarrow S$ with $I \subset \tilde{I}$ and $\left.\tilde{\gamma}\right|_{I}=\gamma$ satisfies $I=\tilde{I}$. A surface is said to be geodesically complete if all maximal geodesics are defined on $I=(-\infty, \infty)$, otherwise, the surface is said to be geodesically incomplete. Give an example, with justification, of a non-compact geodesically complete surface $S$ which is not a plane.

(c) Assume that along any maximal geodesic

$$\gamma:\left(-T_{-}, T_{+}\right) \rightarrow S$$

the following holds:

$$\tag{*} T_{\pm}<\infty \Longrightarrow \limsup _{s \rightarrow T_{\pm}}|K(\gamma(\pm s))|=\infty$$

Here $K$ denotes the Gaussian curvature of $S$.

(i) Show that $S$ is inextendible, i.e. if $\widetilde{S} \subset \mathbb{R}^{3}$ is a connected surface with $S \subset \widetilde{S}$, then $\widetilde{S}=S$.

(ii) Give an example of a surface $S$ which is geodesically incomplete and satisfies $(*)$. Do all geodesically incomplete inextendible surfaces satisfy $(*)$ ? Justify your answer.

[You may use facts about geodesics from the course provided they are clearly stated.]