Let

Let

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

and

So the geodesic curvature of a v-curve is

Similarly we have . That is all the u-curves are

geodesics.

Using the Liouville’s formula

for geodesic curvature, we have the equation

,

where is the directed angle between the geodesic and the

v-curve.

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

getting down.

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.