I'm confused by an example given in my notes regarding connection coefficients, it says consider the 2-D sphere there are 8 connections which are
$Γ^θ_{θθ} = Γ^θ_{θφ} = Γ^θ_{φθ} = Γ^φ _{θθ} = Γ^φ_{φφ} = 0$
$Γ^θ_{φφ} = \tfrac{1}{2} (−∂_θ sin^2 θ) = −cosθsinθ$
$Γ^φ _{θφ} = Γ^φ_{ φθ} =\tfrac{1}{2} sin^2 θ (∂_θ sin^2 θ) = cotθ$
I don't know why my lecturer chose to use $φ,\theta$ as variables a more natural choice would seem to me to be $r, \phi$ so it would be easier to distinguish between the variables, so I'm going to use those instead.
The formula for the connection coefficients is :
$$Γ^{\mu}_{\nu p}=\tfrac{1}{2}(g^{-1})^{\mu \lambda}(d_{\nu}g_{\lambda \rho}+d_{\rho}g_{\nu \lambda}-d_{\lambda}g_{\nu \rho})$$
The matrix associated is
$g_{\mu\nu}= \begin{pmatrix} R^2 &0\\0&R^2sin^2\phi \end{pmatrix}$
$g^{-1\mu\nu}= \begin{pmatrix} \tfrac{1}{R^2} &0\\0& \tfrac{1}{R^2sin^2\phi} \end{pmatrix}$
When I try to work out $Γ^r_{φφ}$ though this is what I get
$Γ^r_{\phi \phi}=\tfrac{1}{2}(g^{-1})^{r \lambda}(d_{\phi}g_{\lambda \phi}+d_{\phi}g_{\phi \lambda}-d_{\lambda}g_{\phi \phi})=\tfrac{1}{2}(g^{-1})^{r r}(d_{\phi}g_{r \phi}+d_{\phi}g_{\phi r}-d_{r}g_{\phi \phi})+\tfrac{1}{2}(g^{-1})^{r \phi}(d_{\phi}g_{\phi \phi}+d_{\phi}g_{\phi \phi}-d_{\phi}g_{\phi \phi})$
$(g^{-1})^{r\phi}=0$ so
$Γ^r_{\phi \phi}=\tfrac{1}{2}(g^{-1})^{r r}(d_{\phi}g_{r \phi}+d_{\phi}g_{\phi r}-d_{r}g_{\phi \phi})$
similarly $g_{r\phi}=g_{\phi r}=0$
so
$Γ^r_{\phi \phi}=\tfrac{1}{2}(g^{-1})^{r r}(-d_{r}g_{\phi \phi})=\tfrac{1}{2R^2}(-d_r(R^2sin^2(\phi))=\tfrac{sin^2\phi}{R}$
So clearly I have some kind of mistake with my variable but I've tried and tried and can't spot what it is, could anyone please point it out to me ?
Your lecturer is talking about the $2$-dimensional sphere (embedded in $\mathbb{R}^3$). So the radius is always constant $1$ and one common set of coordinates to use consists of two angles, namely longitude and latitude like on the earth (measured from $0$ to $2\pi$ instead of degrees on earth).
Using these coordinates I'm not sure whether your matrix for the metric $g$ is correct but I would have to check first to be sure.