<s>[INST] Geodesic on a surface of revolution using Christoffel Symbols. I have the following problem:
For a function $f:[a,b]\rightarrow \mathbb{R}_{>0}$ and for the open set $U=\{(u_1,u_2)\vert\: a<u_1<u_2, 0\leq u_2<2\pi\}$ consider the (local) surface of revolution $M$ obtained as the image of $\sigma:U\rightarrow U'\subset\mathbb{R}^3$ where
\begin{equation}
\sigma(u_1,u_2) = \left(f(u_1)cos(u_2),f(u_1)sin(u_2),u_1\right)
\end{equation}
For a constant $0\leq c<2\pi$, shot that  the meridian curve $\gamma(t)=\sigma(u_1(t),c)$ is a geodesic in $M$.
The way I have tried to solve this as follows:
Recall that $\gamma(t)=\sigma(\gamma_1(t),\gamma_2(t)) = \sigma(u_1(t),c)$ and consider the geodesic equations:
\begin{equation}
\frac{d^2\gamma_k}{dt^2} + \sum_{i,j=1,2}\Gamma_{ij}^k \frac{d\gamma_i}{dt}\frac{d\gamma_j}{dt} = \frac{d^2\gamma_k}{dt^2} + \Gamma_{1,1}^k\left(\frac{d\gamma_1}{dt}\right)^2 + 2\Gamma_{1,2}^k \frac{d\gamma_1}{dt}\frac{d\gamma_2}{dt} + \Gamma_{2,2}^k\left(\frac{d\gamma_2}{dt}\right)^2 = 0
\end{equation}
which should be satisfied for $k=1,2$ if $\gamma(t)$ is a geodesic ($\Gamma_{ij}^k$ are the Christoffel Symbols). Since $\frac{d\gamma_2}{dt} = 0$, the equations above reduce to
\begin{equation}
\frac{d^2\gamma_k}{dt^2} + \Gamma_{1,1}^k\left(\frac{d\gamma_1}{dt}\right)^2 = 0
\end{equation}
Computing $\Gamma_{1,1}^1$ we get:
\begin{equation}
\Gamma_{1,1}^1 =\frac{1}{2}\left(g^{-1}\right)^{1,1}\frac{\partial g_{1,1}}{\partial u_1}= \frac{\frac{\partial f(u_1)}{\partial u_1}\frac{\partial^2f(u_1)}{\partial u_1^2}}{\left(\frac{\partial f(u_1)}{\partial u_1}\right)^2 + 1}
\end{equation}
and $\Gamma_{1,1}^2 = 0$.
The first fundamental form is:
\begin{equation}
(g_{ij}) =  \begin{pmatrix} 
                 f'(u_1)^2+1 & 0 \\
                 0    & f(u_1)^2 
             \end{pmatrix}       =  \begin{pmatrix} 
                            \frac{1}{f'(u_1)^2+1} & 0 \\
                            0    & \frac{1}{f(u_1)^2} 
                        \end{pmatrix}^{-1}.
\end{equation}
This does not seem to work out, and I was wondering why? What mistake have I made?
Cheers!
 [/INST] Thank you THW, although I ended up doing it another way:
First of all, lets compute the first fundemental form and its inverse:
        \begin{equation}
   (g_{ij}) =  \begin{pmatrix} 
        e & f \\
        f  & g 
      \end{pmatrix}   =  \begin{pmatrix} 
                 f'(u_1)^2+1 & 0 \\
                 0    & f(u_1)^2 
             \end{pmatrix}       =  \begin{pmatrix} 
                            \frac{1}{f'(u_1)^2+1} & 0 \\
                            0    & \frac{1}{f(u_1)^2} 
                        \end{pmatrix}^{-1}.
  \end{equation}
        For $\gamma(t)$ to a geodesic, we only need to show that the geodesic curvature of $\gamma(t)$ is zero. Beltrami’s formula for geodesic curvature states that the geodesic curvature $\kappa_g$ can expressed as:
        \begin{align*}
   \kappa_g &= [\Gamma_{1,1}^2\left(\frac{d\gamma_1}{dt}\right)^3 + \left(2\Gamma_{1,2}^2 - \Gamma_{1,1}^1\right)\left(\frac{d\gamma_1}{dt}\right)^2\frac{d\gamma_2}{dt} + \left(\Gamma_{2,2}^2 - 
       2\Gamma_{1,2}^1\right)\frac{d\gamma_1}{dt}\left({d\gamma_2}{dt}\right)^2  \\
      &\quad\quad\quad - \Gamma_{2,2}^1 \left(\frac{d\gamma_2}{dt}\right)^3 + \frac{d\gamma_1}{dt}\frac{d^2\gamma_2}{dt^2} - \frac{d^2\gamma_1}{dt^2}\frac{d\gamma_1}{dt}]\sqrt{eg-f^2}
  \end{align*}
        Since $\frac{d\gamma_2}{dt} = 0$, the expression reduces to
        \begin{equation}
   \Gamma_{1,1}^2\left(\frac{du_1}{dt}\right)^3\sqrt{eg}
  \end{equation}
        Computing $\Gamma_{1,1}^2$:
        \begin{equation}
   \Gamma_{1,1}^2 = \left(g^{-1}\right)^{2,2}\left(\frac{\partial g_{1,2}}{\partial u_1} + \frac{\partial g_{2,1}}{\partial u_1} - \frac{\partial g_{2,2}}{\partial u_2}\right) = 0
  \end{equation}
        Thus the geodiesic curvature $\kappa_g$ of $\gamma(t)$ is zero and $\gamma(t)$ is therefore a geodesic.
</s>