Derivada covariante de armónicos esféricos

Desde$f = f(\theta,\phi) = Y^l_m(\theta,\phi)$es una función escalar,$\nabla_a f$es un vector covariante, por lo que su derivada covariante es la derivada covariante de un vector covariante, por lo que tenemos

$$\begin{align} \nabla^3 \nabla_3 f &= g^{33} \nabla_3 \nabla_3 f \\ &= g^{33} (\partial_3 \nabla_3 f - \Gamma_{33}^a \nabla_a f) = g^{33} (\partial_3 \partial_3 f - \Gamma_{33}^2 \nabla_2 f) \\ &= g^{33} (\partial_3 \partial_3 f - \Gamma_{33}^2 \partial_2 f) = \frac{1}{\sin^2 \theta} \Bigl[\partial_3 \partial_3 f - (- \sin \theta \cos \theta) \partial_2 f\Bigr] \\ &= \left[\frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} + \frac{\cos \theta}{\sin \theta} \frac{\partial}{\partial \theta}\right] f \end{align}$$ y $$\begin{align} \nabla_3 \nabla_2 f &= \partial_3 \nabla_2 f - \Gamma^a_{32} \nabla_a f = \partial_3 \partial_2 f - \Gamma^3_{32} \partial_3 f = \Biggl[\frac{\partial}{\partial \phi} \frac{\partial}{\partial \theta} - \frac{\cos \theta}{\sin \theta} \frac{\partial}{\partial \phi}\Biggr] f \end{align}$$ y $$\nabla_2 \nabla_2 f = \partial_2 \nabla_2 f - \Gamma_{22}^a \nabla_a f = \frac{\partial^2}{\partial \theta^2} f.$$