I am having some anxiety about my computations here! Given a unit-speed embedding $\gamma(t) = (a(t), b(t))$ of an open interval into the left half-plane, define its surface of revolution in the typical way, using local coordinates $(t, \theta) \mapsto (a(t)\cos{\theta}, a(t)\sin{\theta}, b(t))$. The problem is to compute the metric induced from $\mathbb{R}^3$, compute the Christoffel symbols, check that meridians are geodesics and prove that a circle of latitude is a geodesic iff $a(t)$ takes an extreme value there. I had trouble reading the answers on other threads since they used some equations we don't have.
I compute the metric as $g = dt^2 + a(t)^2d\theta^2$, since $\gamma$ is unit-speed. To compute the Christoffel symbols, we can use $\Gamma_{ij}^k = \frac{1}{2}g^{kl}(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})$, where $g^{kl}$ is the inverse matrix for $g_{kl}$.
I get $$\Gamma_{11}^1 = \Gamma_{12}^2 = \Gamma_{21}^1 = 0, \hspace{1cm} \Gamma_{22}^1 = -a'(t)a(t)$$
$$\Gamma_{11}^2 = \Gamma_{22}^1 = 0 \hspace{1cm} \Gamma_{12}^2 = \Gamma_{21}^2 = \frac{a'(t)}{a(t)}$$
For the meridians I can do it in a sentence using reflection, but I think the idea is to use the geodesic equation, and this won't help me do the last part.
So consider a meridian $\mu(t) = (a(t)\cos\omega_0, a(t)\sin\omega_0, b(t))$ for some fixed $\omega_0$. Then my problem is very pathetically basic: I don't know how to check the geodesic equation on $\mu$. The equation we are supposed to use is $\nabla_{\dot{\mu}}\dot{\mu} = \ddot\mu^k + \dot\mu^i\dot\mu^j\Gamma_{ij}^k = 0$ for each $k$. Now when I try to verify these they come out wrong, so I am missing some change of variable or something really basic. Can somebody show me how to write the $\ddot\mu^k + \dot\mu^i\dot\mu^j\Gamma_{ij}^k$ correctly? Or just for arbitrary curve $\mu$ on the surface, how to write $\dot{\mu}^i$ in the 'right' variables?