Finding incomplete geodesics I have a problem with the notion of incomplete geodesics. Can someone give me a minimal example for such a geodesic?
In particular, I am trying to solve the following exercise:
Consider the upper half plane $\mathbb{H}:= \{(x,y):y>0\}\subset \mathbb{R}$ equiped with the metric: $g_{q}:=\frac{1}{y^q}\delta_i^j$ for some real number $q>0,q\not=2$ (for $q=2$ you obtain the poincare half-plane model.)
Show that $(\mathbb{H},g_q)$ is not geodesically complete. 
(Hint: Consider unit-speed geodesics starting at $(x_0,y_0):=(0,1)$)
So far I found the following system of ordinary differential equations:
$x''\cdot y=q\cdot (x'y')$
$y''\cdot y=\frac{q}{2}((y')^2-(x')^2)$
$y^q=(x')^2+(y')^2$

Hint: Try integrating up (if $q > 2$) or down (if $q < 2$) the $y$-axis.
