Given a warped product, I want to compute the ditance of any two points. First get the equation for the geodesic, then compute the length of the geodesic.

Consider the two-dimensional surface $$ g=dr^2+e^{2r}d\theta^2, \quad r\in (-\infty,\infty), \quad \theta \in [0,2\pi] $$ Let $s\to (r(s),\theta(s))$ be a geodesic. Using the formula for the geodesic, I can get $$ r''(s)=C^2 e^{-2r} $$ for some constant $C$. But it seems we can't get the explicit form for $C(s)$, so I don't know how to compute the distance.

Given two points $x_1=(r_1,\theta_1), x_2=(r_2,\theta_2)$, is there a formula for $d(x_1,x_2)$ w.r.t $r_1, r_2, \theta_1, \theta_2$?

The second question is : for a warped product $g_M=dr^2+\varphi^2 g_N$, Is there a distance formula for $x_1=(r_1,y_1), x_2=(r_2,y_2)$ w.r.t. $r_1, r_2, d_N(y_1,y_2)$?