Let $\mathcal{M}$ be a two-dimensional manifold with metric $\boldsymbol{g}$ of signature $-+$.

(i) Let $p \in \mathcal{M}$. Use normal coordinates at the point $p$ to show that one can choose two null vectors $\mathbf{V}, \mathbf{W}$ that form a basis of the vector space $\mathcal{T}_{p}(\mathcal{M})$.

(ii) Consider the interval $I \subset \mathbb{R}$. Let $\gamma: I \rightarrow \mathcal{M}$ be a null curve through $p$ and $\mathbf{U} \neq 0$ be the tangent vector to $\gamma$ at $p$. Show that the vector $\mathbf{U}$ is either parallel to $\mathbf{V}$ or parallel to $\mathbf{W}$.

(iii) Show that every null curve in $\mathcal{M}$ is a null geodesic.

[Hint: You may wish to consider the acceleration $a^{\alpha}=U^{\beta} \nabla_{\beta} U^{\alpha}$.]

(iv) By providing an example, show that not every null curve in four-dimensional Minkowski spacetime is a null geodesic.