Problem
Problem 6.53 Consider $M=\mathbb{R}^{2} \backslash\{(0,0)\}$ with the usual metric $g=\mathrm{d} x^{2}+\mathrm{d} y^{2}$ and consider the distance function $d_{g}$ given by
$$
\begin{aligned}
d_{g}: M \times M & \rightarrow \mathbb{R}^{+} \\
(p, q) & \mapsto d_{g}(p, q)=\inf \int_{0}^{1} \sqrt{g\left(\gamma^{\prime}(t), \gamma^{\prime}(t)\right)} \mathrm{d} t,
\end{aligned}
$$
where $\gamma$ denotes a piecewise $C^{\infty}$ curve with $\gamma(0)=p$ and $\gamma(1)=q$.
(i) Compute the distance between $p=(-1,0)$ and $q=(1,0)$.
(ii) Is there a geodesic minimizing the distance between $p$ and $q$ ?
(iii) Is the topological metric space $\left(M, d_{g}\right)$ complete?
(iv) A Riemannian manifold is said to be geodesically complete if every geodesic $\gamma(t)$ is defined for every real value of the parameter $t$. Is in the present case $M$ geodesically complete?
(v) Find an open neighbourhood $U_{p}$ for each point $p \in M$, such that for all $q \in U_{p}$, the distance $d_{g}(p, q)$ be achieved by a geodesic.