Is it true in general that $d((x,y),(x',y'))^2 = d_M(x,x')^2+d_N(y,y')^2$ in a Riemannian Product $M\times N$?
I read (https://math.stackexchange.com/q/2084841) that this holds if in $M$ geodesics are unique. But what is then wrong with this argument: Let $E(\gamma)=\int_0^1|\dot{\gamma}|^2dt$ be the energy of the curve $\gamma: [0,1]\to M$ in a smooth manifold $M$. In a Riemannian product manifold $M\times N$ every curve has the form $\gamma = (\gamma_1,\gamma_2)$ with energy $E((\gamma_1,\gamma_2)) = E(\gamma_1)+E(\gamma_2)$. Then $$d((x,y),(x',y'))^2 = \inf E((\gamma_1,\gamma_2)) = \inf E(\gamma_1) + \inf E(\gamma_2) = d_M(x,x')^2 + d_N(y,y')^2$$ where the infima are taken over all piecewise differentiable curves in $M\times N$ and $M$ respectively $N$.