Christoffel symbols of $S^n$ in polar coordinates

Question

Consider the usual local polar coordinates $\theta_1, \theta_2,..., \theta_n$ on $S^n$. We were taught about Christoffel symbols today and I am trying to see what the Christoffel symbols of $S^n$ would be in these coordinates. I would really appreciate some help on this. Thanks!

Answer 1

The Christoffel symbols here are, I assume, the coefficients of the Levi-Civita connection in coordinates. It's a standard formula (you can look it up online and it should also be derived in the relevant chapter of your textbook) that if you take coordinates $\{x_i\}$, with coordinate vector fields $\{\partial_i\}$, then $$ \Gamma^k_{ij} = \frac{1}{2}\sum_{l,m}g^{im}\bigg( \frac{\partial g_{mk}}{\partial x_l} + \frac{\partial g_{ml}}{\partial x_k} - \frac{\partial g_{kl}}{\partial x_m} \bigg). $$

So you need to know the coefficients of the Riemannian metric, $g_{ij}$. Recall that if the metric is the bilinear form $g$, its local coordinate coefficients are $g_{ij} = g(\partial_i,\partial_j)$. Now we have to go from abstract nonsense to using something concrete about our manifold, $S^n$, and our choice of coordinates.

We know that $S^n$ is a isometrically embedded into $\Bbb{E}^{n+1}$, which means that its metric can be identified with the restriction of the Euclidean metric to the tangent bundle. Which is to say, $g_{S^n}(\partial_i,\partial_j) = g_{\Bbb{E}^{n+1}}(\partial_i,\partial_j)$. Now recall that the coordinate functions $\theta_i$ are the angular directions in polar coordinates ... can you take it from here?