Help with a problem set question?
Consider a 3-dimensional space given by the set of points $\{(x,y,z),x \in R, y \in R, z > 0\}$ with the metric $ds2 = a/z^2(dx^2 + dy^2 + dz^2)$.
b) Consider two geodesic trajectories with initial conditions $v(0) = (0, 0, z), dv/dt(0) = (1, 0, 0)$ and $w(0) = (0, 0, Z + c), dw/dt(0) = (1, 0, 0).$ Describe the geodesic trajectories and see that to leading order in they satisfy the geodesic deviation equation.
The previous part of the problem was to compute the connection and Riemann curvature tensor so I have that. Geodesic trajectories of the Poincare half plane are semi circles, but how do i extend that knowledge and what does the deviation equation have to do with it?
it seems that you can use the definition of its fundamental form :
$ds^{2}=Edudu+Fdudv+Gdvdv$
where, $E=r_{u}r_{u}$ $,$ $F=r_{u}r_{v}$ $,$ $G=r_{v}r_{u}$
is it helpful ?