Then are an othonormal frame on the torus.
Let be the differential operator of ambient space
, D be the Levi Civita connection of the torus, then the
result of D is the projection of the result of to the
tangent plane. Since we have
It's easy to verify that
So the geodesic curvature of a v-curve is
Similarly we have . That is all the u-curves are
Using the Liouville’s formula
for geodesic curvature, we have the equation
where is the directed angle between the geodesic and the
When s=0, since the geodesic is tangent to the
top circle. We have . So
near the starting point will be negative. This shows that
the geodesic turns right( clock-wise) according to the v-curve. So it's
Also note that will never become
otherwise since the u-curves are geodesics, according to the uniqueness of
geodesics, there is a contradiction. So is always
positive. Before we reach the middle circle defined by u=0, keeps decreasing. When we meet u=0
achieves its minimum. After we cross u=0 increases but still less than 0 so the geodesic keeps going down.
Now notice that the torus is a geodesic symmetric space when the base point
is on the middle circle( since we can always turn 180 degree according to
its diameter and get it back), so the situation after we cross the u=0 line
will be symmetric before we reach that.