Geodesics of $\mathbb{S}^{2}\times \mathbb{R}$

Question

There exists a way for calculate the geodesics of $\mathbb{S}^{2} \times \mathbb{R}$, without to use the extensive calculus Of Christoffel symbols and solve the systems of ODES? Someone references are appreciated!

Thanks

Answer 1

You don't need to do any calculation at all.

Since $M=\mathbb{S}^2\times\mathbb{E}^1$ has the product metric, the Levi-Civita connection behaves as you expect by adding what you get from $\mathbb{S}^2$ and $\mathbb{E}^1$. So a geodesic in $M$ is just a geodesic in $\mathbb{S}^2$ and $\mathbb{E}^1$ separately when projected (here geodesic means $\nabla_{\dot\gamma}\dot\gamma=0$, i.e. constant-speed is assumed). And you know the geodesics in the model space $\mathbb{S}^2$ and $\mathbb{E}^1$.