let (M,g) be an riemanian manifold, where $g= \frac{1}{(x^2)^2} ((dx^1)^2 + (dx^2)^2)$ and $M=\{(x^1,x^2)|x^2>0\}$. Consider in M the unparametrized vertical lines $N= \{(a,x^2)|x^2 >0\}, a\in\mathbb R$.
Find parametrization by arc-length for the vertical lines with initial condition $c(0)=(a,1)$. An obvious parametrization of this curve is
$c(t) = (a,1) + (0,tx^2)$
So, computing the length of this curve in M, we get
$L_c (t) = \int_0^t ||\dot{c}(\gamma)|| d\tau = \int_0^t g(\dot{c}(\gamma),\dot{c}(\gamma)) d\tau $
Now im not sure how to solve $g(\dot{c}(\gamma),\dot{c}(\gamma))$. I found two different definitions
$ g(\dot{c}(\gamma),\dot{c}(\gamma)) = \frac{\sqrt{(\dot{c}^1) + (\dot{c}^2)}}{x^2} \qquad (1)$
$ g(\dot{c}(\gamma),\dot{c}(\gamma)) = \frac{\sqrt{(\dot{c}^1) + (\dot{c}^2)}}{x^2 \circ c(t)} \qquad (2)$
Wich one is correct?