Let $X(u,v)$ be a parametrization of a surface $P$. Let $\gamma : s\rightarrow X(u(s),v(s))$ be a curve on this surface.
$\gamma$ is a geodesic on $P$ if and only if :
$u''+(u')^2 \Gamma_{11}^1+2u'v' \Gamma_{12}^1+(v')^2\Gamma_{22}^1=0$
and
$v''+(u')^2\Gamma_{11}^2+2u'v'\Gamma_{12}^2+(v')^2\Gamma_{22}^2=0$
Where $\Gamma$ symbols refers to Christoffel Symbolism.
Let's consider the particular example :
X: $ \left[0,2\pi \right[\times \left]-2,2\right[\rightarrow \mathbb{R^3},(\theta,r)\rightarrow(cosh(r)cosh(\theta),cosh(r)sin(\theta),r)$ , let's say it's a parametrization of a surface P.
Let $\gamma$ be a curve on this surface , $\gamma: s \rightarrow (cosh(r(s))cosh(\theta(s)),cosh(r(s))sin(\theta(s)),r(s))$.
(First question , where would i have to define this curve , on which interval ? I'm confused)
Applying the Geodesic equations , it leads me to the differential system :
$\theta''+2\theta'r' \frac{sinh(r)}{2cosh(r)}=0$
$r''-(\theta')^2 \frac{sinh(r)}{2cosh^2(r)}+(r')^2sinh(r)=0$
(I did my best to make sure the calculations were correct so I think they are)
Assertion 1 : If p is a point of $P$ , given by (a,b,c) for example , then the constant curve $\gamma : s\rightarrow (a,b,c$) is a geodesic of P passing by (a,b,c).
Assertion 2: On P , as it's a revolution surface , all curves of the form $\gamma:s\rightarrow(cosh(c)\times cos(\theta),cosh(c)\times sin(\theta),c)$ where c is a constant and $\gamma$ has image on P , is a geodesic.
Are these Assertions correct ?
Let's look for Geodesic passing by the point (1,0,0) of P , which corresponds to $X(0,0)$ , we can firstly say that $\gamma_1 :s\rightarrow (1,0,0)$ is a first geodesic passing by this point , and we can find a second one for example using assertion 2 : We can say z(s), is constant 0 , which leads to the differential system to be :
$\theta''=0,\theta(0)=0$. It gives us the solution $\theta=c_3\times x$ where $c_3$ is a constant.
We finally have that the curves $\gamma : s\rightarrow (cos(c_3 s),sin(c_3s),0)$ are also geodesic at this point.
I'd be grateful for corrections , of any kind.