I have to show the following:
Let $(X_1,d_1),(X_2,d_2)$ be metric spaces and $X=X_1 \times X_2$ be the space equiped with the product metric $(d_1^2 + d_2^2)^{1/2}$. If $\sigma:I \to X$ is a geodesic, then there exist $c,s \in \mathbb{R}_+$ and geodesics $\sigma_1,\sigma_2$ such that for all $t \in I$ $$\sigma(t)=(\sigma_1(ct),\sigma_2(st))$$ I am really stuck here. I have only been able to show that $c^2 +s^2=1$ which does not help me at all. Thanks in advance.
Hint: If $\sigma$ is a geodesic from $p = \sigma(0)$ to $q = \sigma(1)$, then consider $\pi_i\sigma$ where $\pi_i : X \rightarrow X_i$ is the projection. If $\sigma_i$ is a geodesic $\pi_i(p)$ to $\pi_i(q)$, what can you say about the relationship between the $\pi_i\sigma$ and the $\sigma_i$?