There is a surface $$p(u,v)=(\rm sech \,v \cos{u},\rm sech\,v \sin{u},v-\rm tanh\,v) (v>0)$$
On the surface, there is a curve $$\hat r(t) =p \circ r(t)\,\,,\,\, r(t)=(u(t),v(t))$$
And $$u(t)^2+{\rm cosh}^2v(t)=a^2$$ ($a$ is a constant greater than 1)
In this situation, I want to show that $\hat r$ is a pre-geodesic.
I learned that to show it is a pre-geodesic, I have to show that det(${d\hat r \over dt}, {d^2\hat r \over dt^2}, \nu$) is zero.
I tried but it was very complicated calculation..Is there anyone who can help me?
Thanks for reading