So I'd like to derive the geodesic equations of a cone which I call $\mathcal{C}$. I believe I've done this correctly but would like a second opinion. $\mathcal{C}$ can be described by taking the line $ z=\beta x \text{ for } \beta>0$ in the (x,z) plane, and then rotating the line about the z-axis. So the cone's 'face' is perpendicular to the z-axis. Here is my attempt.
We need to first parametrise $\mathcal{C}$. I do this by taking the vector $(x,0,z)=(x,0,\beta x)$ and rotating it around the z axis to yield a parametrisation:
$$u(x,\phi)^T= \begin{pmatrix} \cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta &0 \\ 0&0&1 \end{pmatrix} \begin{pmatrix} x \\ 0 \\ \beta x\end{pmatrix}$$ which gives me $$u(x,\phi)=(x\cos \theta,x\sin \theta,\beta x)$$ As my par. of $\mathcal{C}$. Now I derive the metric of this cone as follows. \begin{align} \partial_xu=(\cos \theta, \sin \theta,\beta) && \partial_\phi u=(-x\sin\theta, x\cos \theta,0) \end{align} Hence by letting $x^1=x \text{ and } x^2=\phi$ we have \begin{align} &g_{11}=\langle \partial_xu,\partial_xu\rangle=1+\beta^2 \\ &g_{12}=g_{21}=\langle \partial_xu,\partial_\phi u\rangle=0 \\ &g_{22}=\langle \partial_\phi u,\partial_\phi u\rangle=x^2\end{align} Hence we have that \begin{align}(g_{ij})=\begin{pmatrix}1+\beta^2 &0 \\0&x^2 \end{pmatrix}&& (g^{ij})=\begin{pmatrix}\frac{1}{1+\beta^2} &0 \\0&\frac{1}{x^2} \end{pmatrix}\end{align} Now I calculate the Christoffel symbols via $$\Gamma^k_{ij}=\tfrac12 g^{k\rho} \left(\partial_{i}g_{\rho j} + \partial_{j}g_{\rho i} - \partial_{\rho}g_{ij}\right)$$ Which yields $\Gamma^{1}_{11}=\Gamma^{2}_{22}=\Gamma^{2}_{11}=\Gamma^{1}_{12}=\Gamma^{1}_{21}=0$ and also that \begin{align} \Gamma^{1}_{22}=-\frac{x}{1+\beta^2} && \Gamma^{2}_{12}=\Gamma^{2}_{21}=\frac{1}{x}\end{align} Hence putting all of this into the geodesic equation; $$\ddot{x}^k + \Gamma^{k}_{i j }\dot{x}^i\dot{x}^j = 0$$ Gives us the two equations; \begin{align}\ddot{x} -\frac{x\dot{\phi}^2}{1+\beta^2}=0 && \ddot{\phi}+\frac{2\dot{x}\dot{\phi}}{x}=0 \end{align} Where $(x(t),\phi(t))$ traces out the geodesic on $\mathcal{C}$ for parameter $t$. Do these look correct? - Thanks guys!