# Covariant derivative of spherical harmonics

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: $$$$\nabla_{2}\nabla_{2} Y^l_m = \nabla_{2}\left(\partial_\theta Y^l_m + \Gamma^3_{23}Y^3_m - \Gamma^3_{23}Y^l_3\right)\\ \nabla_{3}\nabla_{3} Y^l_m = ...\\ \nabla_{3}\nabla_{2} Y^l_m = ...,$$$$ where the $$Y^l_m$$ denotes the spherical harmonic functions. In the first line I only used the definition for the covariant derivative and cancel out all Christoffels, which are zero.
The solutions should be: $$$$\nabla_{2}\nabla_{2} Y^l_m = \frac{\partial^2}{\partial\theta^2}Y^l_m\\ \nabla_{3}\nabla_{3} Y^l_m = \left(\frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial\phi^2}+\frac{\cos(\theta)}{\sin(\theta)}\frac{\partial}{\partial\theta}\right)Y^l_m\\ \nabla_{3}\nabla_{2} Y^l_m = \left(\frac{\partial}{\partial\theta}\frac{\partial}{\partial\phi}-\frac{\cos(\theta)}{\sin(\theta)}\frac{\partial}{\partial\phi}\right)Y^l_m.$$$$ Does anyone know, how do get these expressions?

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 $$\nabla_2 \nabla_2 f = \partial_2 \nabla_2 f - \Gamma_{22}^a \nabla_a f = \frac{\partial^2}{\partial \theta^2} f.$$