Vertical lines are geodesics

[odyssey]

For the following Riemannian, hyperbolic metric on the upper half plane \(\displaystyle H = \{(u,v)\in \mathbb{R}^2 \ | \ v>0\},\) how can I prove that the vertical lines are geodesics and that the intersection of any circle centered on \(\displaystyle x-\text{axis}\) with \(\displaystyle H\) is a geodesic?

The definition I have for geodesic is:

Let \(\displaystyle \alpha : [a, b] \to \Sigma\) be a regular parameterized curve then we call it geodesic if its tangent vector is parallel along \(\displaystyle \alpha,\) i.e
\(\displaystyle \nabla_{\alpha '} \alpha'=0.\)

 

[Romsek]

For the following Riemannian, hyperbolic metric on the upper half plane \(\displaystyle H = \{(u,v)\in \mathbb{R}^2 \ | \ v>0\},\) how can I prove that the vertical lines are geodesics and that the intersection of any circle centered on \(\displaystyle x-\text{axis}\) with \(\displaystyle H\) is a geodesic?

The definition I have for geodesic is:

Let \(\displaystyle \alpha : [a, b] \to \Sigma\) be a regular parameterized curve then we call it geodesic if its tangent vector is parallel along \(\displaystyle \alpha,\) i.e
\(\displaystyle \nabla_{\alpha '} \alpha'=0.\)

someone asked this exact question here. There are a few answers there.

 

[odyssey]

Yes, I looked at that answer prior to posting but I didn't understand that answer. I was hoping for a more computational answer rather than that answer you linked.

 

[Romsek]

Yes, I looked at that answer prior to posting but I didn't understand that answer. I was hoping for a more computational answer rather than that answer you linked.

maybe this ?