I want to compute the connection Laplacian for a vector field $\mathbf{T} = T^i \frac{\partial}{\partial x^i}$ on the sphere $S^2$ $$ \text{Lap}(T) = \text{div}(\text{grad} \,T) = \text{tr}_g(\nabla T) = g^{ij} T^k_{i;j}. \tag{1}\label{def} $$
I could only find two papers that give the same explicit expression for it. However, the one I derive doesn't match the one on the literature. What am I doing wrong?
Also, is $\eqref{def}$ the right definition? Is that how one takes $\text{tr}_g$?
My version
In my case, the parametrization of the sphere is given by $$ \mathbf{x}(\theta,\phi) = (\sin(\theta)\cos(\phi), \sin(\theta)\sin(\phi),\cos(\theta)) $$ and I'm using the non-normalized basis vectors $$ \begin{align} &\boldsymbol{\hat{\theta}} = \frac{\partial \mathbf{x}}{\partial \theta} = (\cos(\theta)\cos(\phi), \cos(\theta)\sin(\phi),-\sin(\theta)) \\ % &\boldsymbol{\hat{\phi}} = \frac{\partial \mathbf{x}}{\partial \phi} = (-\sin(\theta)\sin(\phi), \sin(\theta)\cos(\phi),0) \end{align} $$
And I get the following expression (for the calculations, see below): $$ \begin{align} \text{Lap}(T) = &\Bigg[ \Bigg(\frac{\partial^2}{(\partial x^\theta)^2} + \frac{1}{\sin^2(\theta)} \frac{\partial^2}{(\partial x^\phi)^2} + \cot(\theta) \frac{\partial}{\partial x^\theta} - \cot^2(\theta) \Bigg) T^\theta - \Bigg( 2 \, \cot(\theta) \frac{\partial}{\partial x^\phi} \Bigg) T^\phi \Bigg] \boldsymbol{\hat{\theta}} \; + \\ % % &\Bigg[ \Bigg( \frac{\partial^2}{(\partial x^\theta)^2} + \frac{1}{\sin^2(\theta)} \frac{\partial^2}{(\partial x^\phi)^2} + 3 \, \cot(\theta) \frac{\partial}{\partial x^\theta} -1 \Bigg) T^\phi + \Bigg( 2 \, \frac{\cot(\theta)}{\sin^2(\theta)} \frac{\partial}{\partial x^\phi} \Bigg) T^\theta \Bigg] \boldsymbol{\hat{\phi}} \end{align} $$ where I used that $x^\theta = \theta$ and $x^\phi = \phi$.
Literature
The literature reports the following:
First reference: Moon, P. and Spencer, D. E. "The Meaning of the Vector Laplacian." J. Franklin Inst. 256, 551-558, 1953. --> I believe the expression has an error, but on this webpage is written correctly.
Second reference: "Computational Harmonic Analysis for Tensor Fields on the Two-Sphere" J. Comp. Phys. 162, 514–535 (2000) For simplicity I add a picture of the expression at the end.
Details of calculations
Metric is given by: $g_{\theta\theta} = 1$, $g_{\phi\phi} = \sin^2(\theta)$ and $g^{\theta\theta} = 1$, $g^{\phi\phi} = \frac{1}{\sin^2(\theta)}$. The only Christoffel symbols that are non-zero are: $\Gamma^\theta_{\phi\phi} = -\sin(\theta)\cos(\theta)$ and $\Gamma^\phi_{\phi\theta} = \Gamma^\phi_{\theta\phi} = \cot(\theta)$.
$$ \require{cancel} \begin{align} [\text{div}(\nabla T)]^\theta &= g^{ij} T^\theta_{i;j} \\ \nonumber &= g^{ij} \frac{\partial^2 T^\theta}{\partial x^j \partial x^i} + g^{ij} \Gamma^\theta_{pi} \frac{\partial T^p}{\partial x^j} +g^{ij} T^p \frac{\partial \Gamma^\theta_{pi}}{\partial x^j} \\ \nonumber &+ g^{ij} \Gamma^\theta_{mj} \frac{\partial T^m}{\partial x^i} + g^{ij} T^n \, \Gamma^\theta_{mj} \, \Gamma^m_{ni} \\ \nonumber &- g^{ij} \Gamma^l_{ij} \frac{\partial T^\theta}{\partial x^l} - g^{ij} T^p \, \Gamma^l_{ij} \, \Gamma^\theta_{pl} \\ \nonumber % % &= \Big( g^{\theta \theta} \frac{\partial^2 T^\theta}{(\partial x^\theta)^2} + g^{\phi \phi} \frac{\partial^2 T^\theta}{(\partial x^\phi)^2} \Big) + \Big( g^{\phi \phi} \, \Gamma^\theta_{\phi \phi} \frac{\partial T^\phi}{\partial x^\phi} \Big) + \Big( g^{\phi \phi} T^\phi \cancel{\frac{\partial \Gamma^\theta_{\phi \phi}}{\partial x^\phi}} \Big) \\ \nonumber &+ \Big( g^{\phi\phi} \Gamma^\theta_{\phi \phi} \frac{\partial T^\phi}{\partial x^\phi} \Big) + \Big( g^{\phi\phi} T^\theta \, \Gamma^\theta_{\phi \phi} \, \Gamma^\phi_{\theta \phi} \Big) \\ \nonumber &- \Big( g^{\theta \theta} \cancel{\Gamma^l_{\theta \theta}} \frac{\partial T^\theta}{\partial x^l} + g^{\phi \phi} \Gamma^\theta_{\phi \phi} \frac{\partial T^\theta}{\partial x^\theta} \Big) - \Big( \cancel{g^{ij} T^\phi \, \Gamma^\phi_{ij} \, \Gamma^\theta_{\phi \phi}} \Big) \\ \nonumber % % &= g^{\theta \theta} \frac{\partial^2 T^\theta}{(\partial x^\theta)^2} + g^{\phi \phi} \frac{\partial^2 T^\theta}{(\partial x^\phi)^2} + g^{\phi \phi} \, \Gamma^\theta_{\phi \phi} \frac{\partial T^\phi}{\partial x^\phi} \\ \nonumber &+ g^{\phi\phi} \Gamma^\theta_{\phi \phi} \frac{\partial T^\phi}{\partial x^\phi} + g^{\phi\phi} T^\theta \, \Gamma^\theta_{\phi \phi} \, \Gamma^\phi_{\theta \phi} - g^{\phi \phi} \Gamma^\theta_{\phi \phi} \frac{\partial T^\theta}{\partial x^\theta} \\ \nonumber % % &= \Big\{ g^{\theta \theta} \frac{\partial^2}{(\partial x^\theta)^2} + g^{\phi \phi} \frac{\partial^2}{(\partial x^\phi)^2} - g^{\phi \phi} \Gamma^\theta_{\phi \phi} \frac{\partial}{\partial x^\theta} + g^{\phi\phi} \, \Gamma^\theta_{\phi \phi} \, \Gamma^\phi_{\theta \phi} \Big\} T^\theta \\ \nonumber &+ \Big\{ 2 \, g^{\phi\phi} \Gamma^\theta_{\phi \phi} \frac{\partial}{\partial x^\phi} \Big\} T^\phi \\ \nonumber % % &= \Big\{\frac{\partial^2}{(\partial x^\theta)^2} + \frac{1}{\sin^2(\theta)} \frac{\partial^2}{(\partial x^\phi)^2} + \cot(\theta) \frac{\partial}{\partial x^\theta} - \cot^2(\theta) \Big\} T^\theta - \Big\{ 2 \, \cot(\theta) \frac{\partial}{\partial x^\phi} \Big\} T^\phi \\ \nonumber \end{align} $$
and
$$ \begin{align} [\text{div}(\nabla T)]^\phi &= g^{ij} T^\phi_{i;j} \\ \nonumber &= g^{ij} \frac{\partial^2 T^\phi}{\partial x^j \partial x^i} + g^{ij} \Gamma^\phi_{pi} \frac{\partial T^p}{\partial x^j} +g^{ij} T^p \frac{\partial \Gamma^\phi_{pi}}{\partial x^j} \\ \nonumber &+ g^{ij} \Gamma^\phi_{mj} \frac{\partial T^m}{\partial x^i} + g^{ij} T^n \, \Gamma^\phi_{mj} \, \Gamma^m_{ni} \\ \nonumber &- g^{ij} \Gamma^l_{ij} \frac{\partial T^\phi}{\partial x^l} - g^{ij} T^p \, \Gamma^l_{ij} \, \Gamma^\phi_{pl} \\ \nonumber % % &= \Big( g^{\theta \theta} \frac{\partial^2T^\phi}{(\partial x^\theta)^2} + g^{\phi \phi} \frac{\partial^2T^\phi}{(\partial x^\phi)^2} \Big) + \Big( g^{\phi \phi} \, \Gamma^\phi_{\theta\phi} \frac{\partial T^\theta}{\partial x^\phi} + g^{\theta \theta} \, \Gamma^\phi_{\phi\theta} \frac{\partial T^\phi}{\partial x^\theta}\Big) \\ \nonumber &+ \Big( g^{\phi \phi} T^\theta \cancel{\frac{\partial \Gamma^\phi_{\theta \phi}}{\partial x^\phi}} + g^{\theta \theta} T^\phi \frac{\partial \Gamma^\phi_{\phi \theta}}{\partial x^\theta} \Big) + \Big( g^{\phi\phi} \Gamma^\phi_{\theta \phi} \frac{\partial T^\theta}{\partial x^\phi} + g^{\theta\theta} \Gamma^\phi_{\phi \theta} \frac{\partial T^\phi}{\partial x^\theta} \Big) \\ \nonumber &+ \Big( g^{\phi\phi} T^\phi \, \Gamma^\phi_{\theta \phi} \, \Gamma^\theta_{\phi\phi} + g^{\theta\theta} T^\phi \, \Gamma^\phi_{\phi \theta} \, \Gamma^\phi_{\phi\theta} \Big) - \Big( g^{\theta\theta} \cancel{\Gamma^l_{\theta\theta}} \frac{\partial T^\phi}{\partial x^l} + g^{\phi\phi} \Gamma^\theta_{\phi\phi} \frac{\partial T^\phi}{\partial x^\theta} \Big) \\ \nonumber &- \Big( \cancel{g^{ij} T^\theta \, \Gamma^\phi_{ij} \, \Gamma^\phi_{\theta \phi}} + g^{\phi\phi} T^\phi \, \Gamma^\theta_{\phi\phi} \, \Gamma^\phi_{\phi \theta} \Big) \\ \nonumber % % &= g^{\theta \theta} \frac{\partial^2T^\phi}{(\partial x^\theta)^2} + g^{\phi \phi} \frac{\partial^2T^\phi}{(\partial x^\phi)^2} + g^{\phi \phi} \, \Gamma^\phi_{\theta\phi} \frac{\partial T^\theta}{\partial x^\phi} + g^{\theta \theta} \, \Gamma^\phi_{\phi\theta} \frac{\partial T^\phi}{\partial x^\theta} \\ \nonumber &+ g^{\theta \theta} T^\phi \frac{\partial \Gamma^\phi_{\phi \theta}}{\partial x^\theta} + g^{\phi\phi} \Gamma^\phi_{\theta \phi} \frac{\partial T^\theta}{\partial x^\phi} + g^{\theta\theta} \Gamma^\phi_{\phi \theta} \frac{\partial T^\phi}{\partial x^\theta} \\ \nonumber &+ \cancel{g^{\phi\phi} T^\phi \, \Gamma^\phi_{\theta \phi} \, \Gamma^\theta_{\phi\phi}} + g^{\theta\theta} T^\phi \, \Gamma^\phi_{\phi \theta} \, \Gamma^\phi_{\phi\theta} - g^{\phi\phi} \Gamma^\theta_{\phi\phi} \frac{\partial T^\phi}{\partial x^\theta} - \cancel{g^{\phi\phi} T^\phi \, \Gamma^\theta_{\phi\phi} \, \Gamma^\phi_{\phi \theta}} \\ \nonumber % % &= \Big\{ g^{\theta \theta} \frac{\partial^2}{(\partial x^\theta)^2} + g^{\phi \phi} \frac{\partial^2}{(\partial x^\phi)^2} + 2 \, g^{\theta \theta} \, \Gamma^\phi_{\phi\theta} \frac{\partial}{\partial x^\theta} + g^{\theta \theta} \frac{\partial \Gamma^\phi_{\phi \theta}}{\partial x^\theta} \\ \nonumber &+ g^{\theta\theta} \, (\Gamma^\phi_{\phi \theta})^2 - g^{\phi\phi} \Gamma^\theta_{\phi\phi} \frac{\partial}{\partial x^\theta}\Big\} T^\phi + \Big\{ 2 \, g^{\phi \phi} \, \Gamma^\phi_{\theta\phi} \frac{\partial}{\partial x^\phi} \Big\} T^\theta \\ \nonumber % % &= \Big\{ \frac{\partial^2}{(\partial x^\theta)^2} + \frac{1}{\sin^2(\theta)} \frac{\partial^2}{(\partial x^\phi)^2} + 3 \, \cot(\theta) \frac{\partial}{\partial x^\theta} -1 \Big\} T^\phi + \Big\{ 2 \, \frac{\cot(\theta)}{\sin^2(\theta)} \frac{\partial}{\partial x^\phi} \Big\} T^\theta \end{align} $$