I am looking for a way to write down the Christoffel symbols for a surface of revolution.
But I want to use them for a seminar, so I would rather prefer a compressed equation that contains the simplification of the symbols for this particular surface. Thus, I look for an equation that is simpler than the general definition, but something that is more straightforward than all these explicit representations.
Let $x:p\mapsto(x^1(p),x^2(p))$ be the identity chart on $\mathbb{R}^2$ and let $x:p\mapsto(x^1(p),x^2(p),x^3(p))$ be the identity chart on $\mathbb{R}^3$. Suppose $\gamma:t\mapsto(\gamma^1(t),\gamma^2(t))$ is a (smooth) curve in $\mathbb{R}^2$ such that $(\partial_t\gamma^1)^2+(\partial_t\gamma^2)^2\neq0$ and $\gamma^1(t)\neq0$ for each $t$ in the domain of $\gamma$.
The image of the map $$f:U\to\mathbb{R}^3,~(t,\theta)\mapsto(\gamma^1(t)\cos\theta,\gamma^1(t)\sin\theta,\gamma^2(t))$$ is the surface of revolution from the curve $\gamma$.
The induced metric $\mathrm{g}=f^*\left(\sum_{i=1}^3\mathrm{d}x^i\otimes\mathrm{d}x^i\right)$ is \begin{align*}\sum_{i}f^*(\mathrm{d}x^i\otimes\mathrm{d}x^i) & =\sum_i\mathrm{d}f^i\otimes\mathrm{d}f^i \\ & =\sum_i\left((\partial_tf^i)^2\mathrm{d}t\otimes\mathrm{d}t+(\partial_\theta f^i)^2\mathrm{d}\theta\otimes\mathrm{d}\theta\right. \\ & \hspace{4em}\left.+(\partial_tf^i)(\partial_\theta f^i)(\mathrm{d}\theta\otimes\mathrm{d}t+\mathrm{d}t\otimes\mathrm{d}\theta)\right) \\ & =\sum_i\left((\partial_tf^i)^2\mathrm{d}t\otimes\mathrm{d}t+(\partial_\theta f^i)^2\mathrm{d}\theta\otimes\mathrm{d}\theta\right).\end{align*}
In this case, $\mathrm{g}=\!\!\left((\partial_t\gamma^1)^2+(\partial_t\gamma^2)^2\right)\mathrm{d}t\otimes\mathrm{d}t+(\gamma^1)^2\mathrm{d}\theta\otimes\mathrm{d}\theta$. Using our coordinate formulae for the Christoffel symbols of the Levi-Civita connection, we get \begin{align*}\Gamma_{11}^1 & =\frac{\partial_t\left((\partial_t\gamma^1)^2+(\partial_t\gamma^2)^2\right)}{2\left((\partial_t\gamma^1)^2+(\partial_t\gamma^2)^2\right)}=\frac{(\partial_t\gamma^1)\partial_t^2\gamma^1+(\partial_t\gamma^2)\partial_t^2\gamma^2}{\left((\partial_t\gamma^1)^2+(\partial_t\gamma^2)^2\right)}, \\ \Gamma_{11}^2 & =0, \\ \Gamma_{12}^1 & =\Gamma_{21}^1=0, \\ \Gamma_{12}^2 & =\Gamma_{21}^2=\frac{\partial_t\gamma^1}{\gamma^1}, \\ \Gamma_{22}^1 & =-\frac{\gamma^1\partial_t\gamma^1}{\left((\partial_t\gamma^1)^2+(\partial_t\gamma^2)^2\right)}, \\ \Gamma_{22}^2 & =0.\end{align*}