Given is the metric $\gamma_{jk}$ for the surface of a Sphere $S^2$ with $\gamma_{22}=1,\gamma_{23}=\gamma_{32}=0$ and $\gamma_{33}=\sin^2(\theta)$. The coordinates are $x=$($t,r,\theta,\phi$) and $j$ and $k$ run over $x^2=\theta$ and $x^3=\phi$.
Therefore I get the Christoffels $\Gamma^2_{33}=-\sin(\theta)\cos(\theta)$ and $\Gamma^3_{23}=\Gamma^3_{32}=\frac{\cos(\theta)}{\sin(\theta)}$. All other entries of the Christoffels are zero.
Now I want to calculate the following second covariant derivatives with respect to this metric:
\begin{equation}
\nabla_{2}\nabla_{2} Y^l_m = \nabla_{2}\left(\partial_\theta Y^l_m + \Gamma^3_{23}Y^3_m - \Gamma^3_{23}Y^l_3\right)\\
\nabla_{3}\nabla_{3} Y^l_m = ...\\
\nabla_{3}\nabla_{2} Y^l_m = ...,
\end{equation}
where the $Y^l_m$ denotes the spherical harmonic functions. In the first line I only used the definition for the covariant derivative and cancel out all Christoffels, which are zero.
The solutions should be:
\begin{equation}
\nabla_{2}\nabla_{2} Y^l_m = \frac{\partial^2}{\partial\theta^2}Y^l_m\\
\nabla_{3}\nabla_{3} Y^l_m = \left(\frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial\phi^2}+\frac{\cos(\theta)}{\sin(\theta)}\frac{\partial}{\partial\theta}\right)Y^l_m\\
\nabla_{3}\nabla_{2} Y^l_m = \left(\frac{\partial}{\partial\theta}\frac{\partial}{\partial\phi}-\frac{\cos(\theta)}{\sin(\theta)}\frac{\partial}{\partial\phi}\right)Y^l_m.
\end{equation}
Does anyone know, how do get these expressions?
1 Answer
Since $f = f(\theta,\phi) = Y^l_m(\theta,\phi)$ is a scalar function, $\nabla_a f$ is a covariant vector, and so it's covariant derivative is that of the covariant derivative of a covariant vector, thus we have \begin{align} \nabla^3 \nabla_3 f &= g^{33} \nabla_3 \nabla_3 f \\ &= g^{33} (\partial_3 \nabla_3 f - \Gamma_{33}^a \nabla_a f) = g^{33} (\partial_3 \partial_3 f - \Gamma_{33}^2 \nabla_2 f) \\ &= g^{33} (\partial_3 \partial_3 f - \Gamma_{33}^2 \partial_2 f) = \frac{1}{\sin^2 \theta} \Bigl[\partial_3 \partial_3 f - (- \sin \theta \cos \theta) \partial_2 f\Bigr] \\ &= \left[\frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} + \frac{\cos \theta}{\sin \theta} \frac{\partial}{\partial \theta}\right] f \end{align} and \begin{align} \nabla_3 \nabla_2 f &= \partial_3 \nabla_2 f - \Gamma^a_{32} \nabla_a f = \partial_3 \partial_2 f - \Gamma^3_{32} \partial_3 f = \Biggl[\frac{\partial}{\partial \phi} \frac{\partial}{\partial \theta} - \frac{\cos \theta}{\sin \theta} \frac{\partial}{\partial \phi}\Biggr] f \end{align} and $$ \nabla_2 \nabla_2 f = \partial_2 \nabla_2 f - \Gamma_{22}^a \nabla_a f = \frac{\partial^2}{\partial \theta^2} f.$$
