1
  • What's the expression of the Levi-Civita connection of the circle with its metric $d\theta^2$?

More generally,

  • What's the expression of the Levi-Civita connection of the sphere of dimension $n$ with its metric ${r^2} \, d\phi_{1}^2 + {r^2} \sum_{a=2}^{n} \left( \prod_{m=1}^{a-1} \sin^2{\phi_{m}} \right) d\phi_{a}^2$?

The circle is both a Lie group and a Riemannian manifold, with a bi-invariant metric. Therefore, by a well-known argument, the expression of the Levi-Civita connection must be $\nabla_X Y = \frac{[X,Y]}{2}$, at least when X and Y are both left-invariant. We have $X(\theta) = x(\theta) \frac{\partial}{\partial \theta}$ and similarly for $Y$. Can someone help me finish the calculation?

What about the Levi-Civita of the sphere of dimension $n$?

Useful links:

Jeyrome Sapin
  • 905
  • 2
    Just use the standard formula for the Levi-Civita connection in terms of the metric: $$ g_{ip}\Gamma^p_{jk} = \frac{1}{2}(\partial_jg_{pk} + \partial_kg_{jp} - \partial_pg_{jk}). $$ – Deane Jun 17 '25 at 20:25

0 Answers0