I'm currently studying Riemannian Geometry and I would like to get familiar with the basic concepts.
I considered the simple Riemannian manifold $(\mathbb{R}^2, can)$ with its Levi-Civita connection $\nabla$. I know that the Christoffel symbols are all $0$ according to the trivial coordinate system but I want to calculate what happen with the polar coordinates.
I want to calculate $$\Gamma_{\rho \rho}^\rho, \quad \Gamma_{\rho \rho}^\theta, \quad \Gamma_{\rho \theta}^\rho, \quad \Gamma_{\rho \theta}^\theta, \quad \Gamma_{\theta \theta}^\rho , \quad \Gamma_{\theta \theta}^\theta. $$ (I don't need to calculate $\Gamma_{\theta \rho}^\rho$ and $\Gamma_{\theta \rho}^\theta$ because of the symmetry of the Levi-Civita connection.
Let's start for example with $\Gamma_{\rho \rho}^\rho$. $$ \Gamma_{\rho \rho}^\rho = d\rho(\nabla_{\partial_\rho}\rho). $$ My idea is to write $\partial_\rho $ with respect to the standard basis $\{\partial_1, \partial_2\}$ and to write $d\rho$ with respect to $\{dx_1, dx_2\}$ and use the properties of $\nabla$.
Therefore: \begin{align} \Gamma_{\rho \rho}^\rho &= d\rho(\nabla_{\partial_\rho}\rho) \\ &=(\cos\theta dx_1 + \sin\theta dx_2)\big(\nabla_{\cos\theta\partial_1 + \sin\theta\partial_2}\cos\theta\partial_1 + \sin\theta\partial_2\big)\\ &\quad\vdots \quad \text{(after some calculations)}\\ &=0. \end{align}
I don't know if the result is correct. I would like to know if this is a correct way to solve the exercise. Are there any other strategies?
Thanks!