In this problem we try to find the critical points (ideally minima) of the energy functional on curves in the hyperbolic plane. Generally, for any Riemannian manifold (U, g) the energy functional is given by $$E(\gamma) = \frac{1}{2} \int_I g_{\gamma (t)} (\gamma '(t), \gamma '(t))dt$$
for curves γ : I → U.
Recall that SL(2, R) acts by isometries on H$^2$ via A · z = $\frac {az + b} {cz + d}$ for A ∈ SL(2, R). Show that the energy does not change when a curve γ is moved by an isometry to A · γ.
So for this I imagine we can just directly compare E($\gamma$) and E($A\gamma$)? I was told that the this equality holds $$g_p(v,w)=g_{A(p)}(D_pA(v),D_pA(w)).$$ But I don't know how exactly to relate that back to my situation. Would it just be something like... $$g_\gamma(t)=g_{A(\gamma)}(A\gamma'(t),A\gamma'(t)).$$
But even then I'm not sure what exactly g$_A(\gamma)$ is.. I mean g$_\gamma$ is the hyperbolic metric: $ds^{2} = \frac{dx^2 + dy^2}{y^2}$ so would g$_A(\gamma)$ be that times A? I'm confused and it someone can show me how to go about this that would be exceptional.
PS. I was also told that, in this context (the upper hyperbolic plane), $E(\gamma) = E(A\gamma)$ is equivalent to $d(z_1,z_2)=d(Az_1,Az_2)$ where $d$ is the hyperbolic distance. Is this true? In this case I could just use the explicit formula for hyperbolic distance with each set of points.
Isometries preserve energy in any reasonable context. Be careful not to lose focus of what matters because of particularities of the hyperbolic plane. If $(M,g)$ and $(N,h)$ are two (Pseudo-)Riemannian manifolds and $\phi\colon M \to N$ is an isometry, that is, $\phi$ is a diffeomorphism that satisfies$$g_p(v_1,v_2) = h_{\phi(p)}({\rm d}\phi_p(v_1),{\rm d}\phi_p(v_2))$$for all $p \in M$, $v_1,v_2 \in T_pM$, then the following holds: if $\alpha\colon I \to M$ is any curve, then $E[\alpha] = E[\phi \circ \alpha]$, where the first energy is computed according to $g$ in $M$, and the second to $h$ in $N$. And the proof is easy: $$\begin{align}E[\alpha] &= \frac{1}{2}\int_I g_{\alpha(t)}(\alpha'(t),\alpha'(t))\,{\rm d}t \\ &= \frac{1}{2}\int_I h_{\phi(\alpha(t))}({\rm d}\phi_{\alpha(t)}(\alpha'(t)),{\rm d}\phi_{\alpha(t)}(\alpha'(t)))\,{\rm d}t\\ &= \frac{1}{2}\int_I h_{(\phi \circ \alpha)(t)}((\phi\circ\alpha)'(t),(\phi \circ \alpha)'(t))\,{\rm d}t \\ &= E[\phi\circ \alpha].\end{align}$$